I made up my mind and here's what I came up with taking into consideration the orthogonality concept. Orthogonality turns out to be a key to understand what's happening when applying Fourier transforms. Then both matrix interpretation suggested by my professor and analyzing integrals depict the idea well.
Let only consider complex pure tone waves of form $e^{2\pi i \xi t}$. It's simpler and more compact than thinking of both sines and cosines.
Recall (as T L Davis kindly pointed out) that the inner product of complex functions $f$ and $g$ from real line is defined as
$$\left<f,g\right>=\int_{-\infty}^{\infty}f(t)\overline{g(t)}\mathrm{d}t$$
It follows that the norm is
$$||f||=\left(\int_{-\infty}^{\infty}|f(t)|^2\mathrm{d}t\right)^{{1}/{2}}$$ which are $L^2$ norm and inner product, that can probably be used to make some formal assumptions for existance of FT for particular functions. Having the above in mind let us step back to Fourier series i.e.
$f(t)=\sum_\limits{n=-\infty}^{\infty}c_n(T)e^{2\pi i\frac{n}{T} t}$
As HBR pointed out within his answer, the coefficients of Fourier series are given by
$$c_n(T)=\frac{1}{T}\int_{-T/2}^{T/2}f(t)e^{-2\pi i \frac{n}{T}t}\mathrm{d}t,$$
but hey, these $c_n(T)$ are exactly what we're asking about. We assume that a periodic function can be decomposed into countably-infinite number of sine waves of given form and we need to know their 'quantitities'. Fourier coefficients are the answer to that problem and Fourier transform is its continuous analogon. So $c_n(T)$ themselves are just a discrete form of FT. By letting $T\to\infty$ and noticing Riemannian sum we get the desired convergence:
$$f(t)=\lim_{T\to\infty}\sum_\limits{n=-\infty}^{\infty}e^{2\pi i\frac{n}{T} t}\int_{-T/2}^{T/2}f(t)e^{-2\pi i \frac{n}{T}t}\mathrm{d}t\frac{1}{T}=$$
$$\int_{-\infty}^{\infty}e^{2\pi i\xi t}\int_{-\infty}^{\infty}f(t)e^{-2\pi i\xi t}\mathrm{d}t\mathrm{d}\xi=\int_{-\infty}^{\infty}F(\xi)e^{2\pi i\xi t}\mathrm{d}\xi.$$
where $F(\xi)$ is the FT. We arrived with the continuous form. Letting $T\to\infty$ can be thought of as considering periodic function with an infinite period, so non-periodic actually.
The only question left is why are the coefficients given by such a formulae. That is where the orthogonality comes in. We know that pure tone waves of different frequencies are orthogonal (from inner product and T L Davis comment). So we can form an orthognal linear basis from solely pure tone waves. It follows that there exists a unique representation of any function in such a basis and the 'quantity' coefficients are given by inner product of considered function with each sine. That's the general idea standing behind the whole thing. Now we have to step back to periodic functions once again. If some function $f$ is periodic on interval $[-\frac{T}{2},\frac{T}{2}]$ then it's sufficient to consider inner product over that interval only.
$$\left<f(t),e^{2\pi i \xi_n t}\right>=\int_{-T/2}^{T/2}f(t)\overline{e^{2\pi i\xi_n t}}\mathrm{d}t=\int_{-T/2}^{T/2}f(t)e^{-2\pi i\xi_n t}\mathrm{d}t,$$
which is nearly exactly what we had for $c_n(T)$ and $\xi_n$ denotes $\frac{n}{T}$. There's still $\frac{1}{T}$ left. Notice that norm of pure tone wave is
$$||e^{2\pi i \xi_n t}||^2=\int_{-T/2}^{T/2}e^{2\pi i \xi_n t}e^{-2\pi i \xi_n t}\mathrm{d}t=\int_{-T/2}^{T/2}1\mathrm{d}t=T.$$
We can see now, that
$$c_n(T) = \frac{\left<f(t),e^{2\pi i \xi_n t}\right>}{||e^{2\pi i \xi_n t}||^2}=\cos(\vartheta_n)\frac{||f(t)||}{||e^{2\pi i \xi_n t}||}$$
Extension to continuous form is then more obvious as $\cos(\cdot)$ is bounded, so we get rid of the infinities when $T\to\infty$.
Knowing that we can understand the integral for Fourier transform as taking inner product of a particular signal with pure tone waves frequency by frequency. Note that FT as defined without division by norm, it informally allows to take infinite values and occurence of delta functions. That explains it enough for me.
With the above we're ready to answer questions stated in my post. Terms cancelletion is now obvious from orthogonality. I've asked about a constant wave. It's orthogonal to all non-cosntant basis function. So
$$
\mathcal{F}\{C\}(\xi)= \begin{cases}
\infty &\mathrm{when} \quad \xi=0 \\
0 &\mathrm{elsewhere} \\
\end{cases}
$$
That goes straightaway from inner product of constant. However, we have to watch out because of limit. We have to distinct between $\delta(k)$ and $C\delta(k)$ they're not the same when though of as limits. The above is $C\delta(k)$ then. What's interesting, if there's a constant component in any singal we can extract it by just integrating this function over the real line.
Focus now on even and odd functions. We notice, that even ones have pure real FT and odd ones pure imaginary. Every function can be easily decomposed into its even and odd part, then we could look at their transforms separately. For example let $f(t)$ be ractantgular pulse on interval $[0,1]$ with amplitude $1$. It's neither odd nor even. Notice that $f(t)=f_e(t)+f_o(t)$, where $f_e(t)$ is recantagular pulse of amplitude $1$ on interval $[-1,1]$ and $f_o(t)$ is $-1$ on $[-1,0]$ and $1$ on $[0,1]$. It can be shown that that first term FT is $\mathrm{sinc}(\xi)$ and second $2i\sin(\xi/2)\mathrm{sinc}(\xi/2)$. They're in fact pure real and pure imaginary, what's more they're even and odd respectively again.
Matrix interpretation is now also easily understable with a bit of linear algebra, but as I see know it's more suitable for discrete Fourier transform, as it regards finite representations. From spectral theorem we know which matrices can be diagonolized. We have
$A=U^{-1}DU$, where U is matrix of transition to an orthogonal basis and D is diagonal. D stands for discrete FT. Most probably it has some interesting properties, but we'll leave it there.
I hope somebody finds this thoughts interesting. Feel free to share your ideas or correct me in case I'm wrong.
