Gaining intuition for what $f(x) = a^{x}$ and $g(x) = x!$ actually means $a^x$ where $x \in \mathbb{Z^{+}}$ is easy to think about.
For example, $2^{3} = \underbrace{2 \times 2 \times 2}_\text{3 times}$.
$a^x$ where $x \in \mathbb{Z^{-}}$ isn't too hard either.
For example, $2^{-3} = \frac{1}{\underbrace{2 \times 2 \times 2}_\text{3 times}}$.
$a^x$ where $x \in \mathbb{Q}$ is still manageable to understand.
For example, $2^{7/3} = 2^{2 + \frac{1}{3}} = \underbrace{2 \times 2}_\text{2 times} \times 2^{\frac{1}{3}} = \underbrace{2 \times 2}_\text{2 times} \times \sqrt[3]{2}$.
However when $x \in \mathbb R \setminus \mathbb Q$, I fail to grasp what $a^{x}$ means.
For example, $2^{\sqrt{2}} = 2^{1 + 0.414...} = \underbrace{2}_\text{1 time} \times 2^{0.414...}$.
But what does it mean to take $0.414...$ to the power of $2$? How can I think about it?
While the above case is hard to grapple with for me, I have absolutely no idea what something like $e^{i\pi}$ actually means. 
How can a number multiply by itself $i\pi$ times when $i$ has no quantitative value other than $\sqrt{-1}$?
On a similar note, I read somewhere that $(\frac{1}{2})! = \frac{\sqrt{\pi}}{2}$. 
Once again, how is this possible, and what does it actually mean?
Perhaps my understanding of exponents and factorials is too simplistic - I understand $a^{x}$ as being $a$ multiplied by itself $x$ times and $n!$ as $1 \times 2 \times 3 \times 4 \times ... \times n$ - or I just need some clarity but either way I would love to gain some intuition as to what is really happening with these functions (maybe some examples would be great!)
 A: I just want to answer the first part of your question.
By definition of $a^x$, you can understand $2^{\sqrt2}$ as follows:
$$2^{\sqrt 2}=2^{1}×2^{0.4}×2^{0.01}×2^{0.004}× \cdots$$
If, $\lim_{n\to\infty} \sum_{k=1}^{n} a_k=\sqrt 2$, then
$$\begin{align}2^{\sqrt 2}=2^{a_1}×2^{a_2}×\cdots × 2^{a_n}×\cdots\end{align}$$
A: For the first question, consider the map $f_k: x\mapsto x^{\sqrt k}$.
Then, applying this map twice simply gives $f_k\circ f_k : x\mapsto x^k$
So, $x^\sqrt2$ can be seen as going half-way, in an exponential sense, to $x^2$

For the second, the answer lies in Euler's Formula: $$re^{i\theta}=r(\cos\theta+i\sin\theta)$$
For any pair $(x,y)\in\Bbb R^2$, it is possible to transpose them into polar co-ordinates as $(r\cos\theta, r\sin\theta)$, using $r=\sqrt{x^2+y^2}$. We can then represent that point $(x,y)\in \Bbb R^2$ on a circle of radius $r$, which has a lot of uses. Our angle $\theta$ determines the rotation anticlockwise from the positive $x$-axis around the circle, and can be capped at $2\pi$ because $\sin$ and $\cos$ are $2\pi$-periodic.
Now, the complex bit. We have: $$\Bbb C=\{x+iy\  |\  x,y\in\Bbb R; i^2=-1\}$$
We may represent any element of $\Bbb C$ by an $(x,y)\in \Bbb R^2$ using the above definition. Then, we may translate that to polar co-ordinates using the work above. So, indeed: $$\Bbb C=\{r\cos\theta+i(r\sin \theta) \ | \ r\in\Bbb R^+_0\, \theta\in[0,2\pi); i^2=-1\}$$
Euler's Formula, as written above, gives that this is also equal to $re^{i\theta}$ (the reason for why is seen in Zadig's excellent answer - input $ix$ into the argument of the series expansion of $e^x$). In this sense, our $r$ represents the size of the circle, and $e^{i\theta}$ represents the rotation of $(r,0)\in\Bbb R^2$ by $\theta$ radians about the origin (i.e. where on the circle the point is).
For $e^{i\pi}$, we have $r=1$ and $\theta=\pi$. Taking the point $(1,0)$ and rotating $\pi$ radians (aka $180^\circ)$ lands you at $(-1,0)\in\Bbb R^2$ which represents $-1\in\Bbb C$. Hence the famous equation $e^{i\pi}+1=0$

On the third question, there are two parts.
Firstly, $n!$ is exactly the number of ways you can order $n$ objects. For example, $ABC$ can be ordered into $ABC, ACB, BAC, BCA, CAB, CBA$, so $6=3!$ configurations. This gives an intuitive representation at least for the integers.
For non-integers, the special property of the factorial function: $n!=n(n-1)!$ provides the reasoning. So forth came the Gamma Function, the extension of the Factorial to non-integers. It is defined by $$\Gamma(z)=\int_{\Bbb R^+}x^{z-1}e^{-x} dx$$
Indeed, by integrating by parts, $$\Gamma(z+1)=\int_{\Bbb R^+}x^{z}e^{-x}dx=[-e^{-x}x^z]|^\infty_0-\int_{\Bbb R^+}zx^{z-1}e^{-x}dx=z\int_{\Bbb R^+} x^{z-1}e^{-x}dx=z\Gamma(z)$$
since the other term in the IBP step vanishes.
In this sense, Gamma extends the notion of factorial (with the shift $\Gamma(z)=(z-1)!$) to non-integer arguments.
We may also evaluate $(\frac12)!=\Gamma(\frac32)$ as $$\Gamma(\frac32)=\frac12\Gamma(\frac12)=\frac12\int_{\Bbb R^+} x^{-\frac12}e^{-x}dx\overbrace{=}^{u=x^\frac12\to \\ 2\ du=x^{-\frac12 }dx}\int_{\Bbb R^+}e^{-u^2}du=\frac{\sqrt\pi}{2}$$
The latter integral is known as the Gaussian Integral.
A: The answer I'll give is very simplistic, but whatever:
You have to understand that not everything is interpreted in terms of numerical values. In the case of exponentiation in $\mathbb{C}$, we know for a fact that for real numbers, the series expansions
$$e^x=\sum_{n\geq 0} \frac{x^n}{n!}=1+x+\frac{x^2}{2!}+ \frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}...$$
$$\cos(x)=\sum_{n\geq 0} \frac{(-1)^n x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...$$
$$\sin(x)=\sum_{n\geq 0} \frac{(-1)^n x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}+...$$
look very similar to one another, which indicates that there must be a link between exponentials and trigonometric functions. That very link is Euler's formula
$$e^{ix}=\cos(x)+i\sin(x)$$
If you try to interpret the meaning of $i$ according to its numerical value, it's purely meaningless. Yet given the equality above, raising a real number to a complex power does have meaning: it returns a complex number on the unit circle in $\mathbb{C}$, since the pair $(\cos(x),\sin(x))$ is on the unit circle.
As for the factorial, we've defined $n!=n(n-1)(n-2)...(2)(1)$ for natural numbers. If you plot the points $(n,n!)$, there are gaps between every pair. What if there existed a function $f:\mathbb{R}^+\to\mathbb{R}$ that managed to correctly return every factorial as defined on $\mathbb{N}$ for integer values, while "closing that gap"? It turns out that such a function exists, that's the gamma function
$$\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}dt=(x-1)!$$
For every integer $x$, it does indeed return $(x-1)! = (x-1)(x-2)...(2)(1)$. The non-integer factorial pairs can be interpreted as simply "being there" so that the curve is smooth.
A: This is a wonderful question, and at the core it is not so much about what $a^x$ means - or $n!$ - but is rather asking, what is a real number, what is a rational number, and what is an imaginary number?  Once we get down to the mysteries of what numbers are, then we start to ponder what numbers mean in Euler's identity and further how rotations of number systems occur in a plane.  The ability to contemplate what numbers are is the force behind becoming an excellent and perceptive mathematician.  First, look closely at the real number line and convince yourself that any real number on the line is essentially the same conceptually, as far as real numbers go. 
Our minds recognize the counting numbers so $2$ may seem to be more of a number than a fractional or irrational number, but every number is represented by one point on the real number line, and a point is described by the intersection of two lines which have no thickness, so a point has no size.  To you, $\sqrt{2}$ might mean more than $(\sqrt{2}-1)$ because when you square the first expression you arrive at a counting number yet when you square the second expression, you arrive at an irrational number $0.17157...$ which is every bit as good as the number $2$, it is just an irrational number rather than a counting number.  While a fraction seems easier to understand than an irrational number, the fraction $\frac13$ does not have a finite decimal representation.  Is a fraction like $\frac12$ more of a real number than $\frac13$ because as a decimal it can be expressed with finite digits as $0.5$?  Is the meaning of a negative fraction like  $-\frac13$ of lesser value, I don't mean quantity-wise but conceptually?  They are all numbers representing one point on the real number line.  A rational number expressed as a fraction could have thousands or millions of digits in the numerator or denominator, and it would still be rational, and would still be a point on the real number line like every other point.  Let's compare $2^{\frac13} \approx 1.255921$ to $2^{\frac{111}{332}} \approx 1.260798$.  It is far preferable to write $\sqrt[3]2$ than $(\sqrt[332]2)^{111}$ I agree but these are very close values.  Our minds might prefer neat simple expressions over excrutiating exactness.
Now, we start to get to the idea of infinity, and have to keep in mind that even when the digits of a number go on to infinity, the point on the real number line still has no size, and each additional decimal is a smaller and smaller fine tuning of the actual value of that number that can not be written out in its entirety.  The counting numbers are infinite, but they are 'countably' infinite, and this is just a definition that means what it was defined to mean.  The even counting numbers do not represent half of infinity.  If X represents a set of the counting numbers, the 2X represents all of the even counting numbers, but each member corresponds to a counting number in X.  Conversely, on the real number line, if you choose any two points, there is an infinite number of points in between them, and the points are not countably infinite.  It seems that there are far more numbers on the real number line that do not have a great deal of meaning than the few that do, even though the meaningful numbers may still be infinite.
So let's look at $x^a$ where $x$ is a real number, whether $x$ be rational or irrational.  Let's consider $x$ as a continuum along the real number line. This plot shows the curves $x^{\frac13}$, $x^{\sqrt{2}-1}$, and $x^{\frac12}$, and the point marked in red is $(2, 2^{\sqrt{2}-1})$.  While $a$ has been set at three constant real numbers, you can imagine $a$ along a continuum of the real number line so that there is a continuum of curves, which aren't very different from one another conceptually even if $(\sqrt{2}-1)$ does not seem like as much of a number as $\frac13$.

We can similarly look at the plots of $a^x$ allowing $x$ to vary along the real number line and choosing fixed values for $a$, and in this case $a$ is set equal to $2, e,$ and $3$.  The point marked in red is $((\sqrt{2}-1), 2^{\sqrt{2}-1})$.  On the graph of $e^x$, we could find $e^{\pi}$ if the line were extended a bit farther.

Once we get into expressions including imaginary numbers, like $e^{i\pi}$ we aren't on the continuum of the real number line any more.  If we go back to the original real number line graph, we can see the real number $i^2  = -1$ is on the line, but $i = \sqrt{-1}$ is not on the line.  It follows that while $2^{2 \times 2 \times 2} = (2^{2 \times 2}) ^ 2 $, it is not the same to say $e^{i\pi} = (e^\pi)^i$.  Consider that an imaginary number can be written as $z = a + bi$, where $a$ is the real part and $bi$ is the imaginary part, then if $z = i\pi, a = 0$ and $b = \pi$.  Perhaps think of $a$ as the coefficient on the real number $'1'$ expressing the magnitude in the direction of the real number line, and $b$ as the coefficient on the imaginary number $'i'$ expressing the magnitude in the direction of the imaginary number line.  It is not quite the same thing as multiplication, although the imaginary number $'i'$ does fall out of the quadratic equation where there are no real solutions, and the coefficient does act in a manner similar to multiplication.  I would refer you to the youtube series on Complex Analysis by Petra Confer-Taylor to gain a deeper understanding of $'i'$ and how it is useful in achieving real solutions, or else meaningful ones.  To add to the answer given by @lone student in the third part, if you go further to explore the steps to the Gaussian Integral, you will find within it a rotation.  Rotations occur in mathematics, in the plane, for many reasons.  You might think of polar coordinates, or the complex plane, or trigonometric functions.  There are cycles of negatives and positives that occur by raising $-1$ to integer powers, and cycles from derivatives of $cosx$ and $sinx$, and the integer powers of imaginary numbers cycling between real and imaginary, positive and negative. The equation of a circle is $x^2 + y^2 = r^2$ and if we allow imaginary solutions then the equation could be $x^2 - y^2 = r^2$.  Then $\pi$ will fall out of equations that are circular, or else from the hypotenuse of a right triangle by the Pythagorean Theorem.  The Gamma Function was defined to fill in the values between integers and create a continuum such as is seen on the real number line.  Factorials of the values intermediate to the integers do not have a meaning other than the equation that has been defined to fill in those points.  I so very much hope that these are the answers that begin to make mathematics and the meaning of numbers come together for you...and please let me know where your questions take you!
