Confused about the $\pm$ sign? I have multiple questions about the $\pm$ sign, since it seems to confuse me in general...
Question 1:
Say I have $15=\pm(a+x)$, Can I use the distributive property so it becomes $15=\pm a \pm x$? Or does that mean I went from 2 to 4 solutions?
I get very confused when I encounter this sign in equations I need to simplify. Whenever I need to deal with these kind of situations I tend to just turn it into two equations, $15= a+x$ and $15=-a-x$. But if I were to do that with bigger equations that still need to be simplified it means I'm wasting alot of time since I'm doing twice the labor...
Question 2:
Does the $\pm$ sign only make sense in equations, or can they be used in normal expressions as well (e.g. $\pm y + 3$). But then I wonder, in what scenario would you do something like this? 
Question 3:
$\sqrt {x^2}+37=y+40$
Say I were to simplify $\sqrt{x^2}$ in that equation, I don't know where I should put the $\pm$ sign. Where would I put it? The extra terms 37 and 40 confuse me...
 A: If the $\pm$ sign is confusing you, get rid of it.  If you have $$15 = \pm(a+x)$$ you can turn that into two equations:  $$15 = a+x\\15 = -(a+x)$$ and then deal with the two equations separately, one at a time.  That is exactly the meaning of the $\pm$ sign.

The reason you're confused is because the notation is confusing!  The expression $\pm a \pm b$ is actually ambiguous: in some contexts it means four values, and in other contexts it means two. Sometimes there is a convention that the two $\pm$ signs must represent the same sign; sometimes there isn't.
In your third example, I'd suggest that you write $z = \sqrt{x^2}$ and turn the equation into $z+37 = y + 40$.  Then solve as usual.  When you get to the end, you have $z$ and $y$.  Then you can conclude that $x$ could be either $z$ or $-z$.
A: 1: Yes, you can distribute it.
Well, it works for all cases.
$$\pm(x + y) = \pm x \pm y \quad \forall x,y \in \mathbb R$$
The basic properties are:
$$(-)\times (\pm) = (\mp)\\
 (\pm)\times(\pm) = (+)\\
 (\mp)\times(\mp) = (+)\\
 (\pm)\times(\mp) = (-)
$$

2:
As you said, $\pm$ is used to represent $+$ and $-$ in separate equations. It's a bit more fundamental than that. Even as a statement

$$x \pm y  \implies x+ y \text{ and } x - y$$
There isn't a real usefulness for it rather than saves time in writing and speaking.
For example,

I ask you, " What is $\sin(A\pm B)$ ?" 
   You'll tell me, " $\sin A \cos B \pm \cos A \sin B$"

I've recently been using it to  help my little brother practice both addition and subtraction simultaneously,
$$3 \pm 2 \to 5, 1\\
32 \pm 23 \to 55, 9\\
1729 \pm 999 \to  2728 , 1630\\
854297992 \pm 299792458 \to 1154090450, 554505534\\
\dots$$
He's been getting really good at it.

3: $\sqrt{x^2} = |x|$

Take the points $\pm x$ on a number line (Hopefully, you understand the usage now)
The distance to those points from the origin can be found by the distance formula:
$$\sqrt{(\pm x - 0)^2} = \sqrt{x^2}$$
Now, $\sqrt{x^2} = x ,\quad\forall x\in [0, \infty)$
and $\sqrt{x^2} = -x ,\quad\forall x \in (-\infty, 0)$
So, it always outputs the magnitude of $x$
Let's try playing with that equation you've given using our knowledge of this:
$$\sqrt {x^2}+37= y + 40 \\
\implies |x| = y + 3 \\
\implies x = y + 3, \space\forall y\in [-3, \infty)\quad\text{ and }\quad x = - (y+3), \space\forall y\in (-\infty, - 3)$$
A: $\bf{Question\ 1:}$ The plus or minus sign does not distribute like that. To see this, consider the equation $x^{2}-4=0$. It should be obvious that the roots of this equation are $\pm 2$. We can rewrite that as $\pm (10-8)$ since $10-8=2$. However, Distributing the $\pm$ sign through $(10-8)$ would imply that $18$ and $-18$ are also roots of $x^{2}-4=0$.
$\bf{Question\ 2:}$ It can be used as you have specified, although it is not very common to see that.
$\bf{Question\ 3:}$ The quantity $\sqrt{x^{2}}$ is used to denote the magnitude of $x$, which is $|x|$.
A: what if variable x = +1 and y = -1, since it is a variable, maybe 4x could be 4+1= 5, not multiplication in this state? y= -1, 2y = 2-1 = 1.
4x -2y=3.
4x-2y=3
_____=3
           _
4-2y=4-2y
x = 3
    ______
     4-2y
3             = +1

4-2y
3
-   = +1
3
x=+1 , y=-1
