Discussion on the Conditions for Pointwise/Uniform Convergence of Fourier Serieses.

Question

My book isn't very clear about the conditions for pointwise/uniform convergence of fourier series; so, after a bit of search, here I am with a summary of what I found. Please, it would be very appreciated if someone could discuss or check the validity of any of these points:

1. Let's suppose that the function $\left[f(x)\right]$ has left-and-right derivatives in the point $\left[x_0\right]$, and is here continuos, then its fourier series converges to $\left[f(x_0)\right]$. Can somebody assure that this statement holds even if such derivatives are not equal?
2. Let's suppose that the function $\left[f(x) \in C^1([-L,+L])\right]$, then its fourier series converges pointwise over the whole segment. Can somebody assure that this convergence is also uniform?
3. Let's suppose that $\left[\sum_{n=-\infty}^{+\infty} c_ne^{inx}\right]$ is the fourier series of the function $\left[f(x)\right]$; if $\left[\sum_{n=-\infty}^{+\infty} |c_n| \lt \infty\right]$ then the series converges absolutely, and therefore uniformly. (NOTE: this is a consequence of m-test.)
4. Let's suppose that the function $\left[f(x)\right]$ has bounded variation over the segment $\left[-\pi, +\pi\right]$, or its first derivative is such that $\left[\;|f'(x)|\le K\;\right]$ (otherwise it is lipshitz, or more generally it is $\alpha$-holder with $0 \le \alpha \le 1$), then its fourier series converges uniformly. Note that this statement holds for $2\pi$-periodic functions (or at least this is what I found). Can somebody assure that this statement holds even over a generic segment $\left[a,b\right]$? Moreover the condition of periodicity has to be contextualized with $\left[\; \lim_{x\to a^+}f(x)=\lim_{x\to b^-}f(x) \;\right]$.
5. Let's suppose that the function $\left[f(x)\right]$ is periodic with bounded variation over a certain segment; then its fourier series converges pointwise to $\left[\lim_{\epsilon\to 0} \frac{f(x+\epsilon)+f(x-\epsilon)}{2}\right]$. In particular, if the function is continuous in $\left[(x)\right]$, then its fourier series converges pointwise to $\left[f(x)\right]$. Moreover, if the function is continuous everywhere over the segment, then its fourier series converges uniformly to $\left[f(x)\right]$. Can somebody discuss about the periodicity condition? Is that necessary?
6. Let's suppose that the function $\left[f(x) \in L^p, \;p\gt1 \right]$; then its fourier series converges for "almost every" $\left[(x)\right]$. Can somebody assure that this type of convergence is a "pointwise-one"?

---

In addition to those, there is a very personal question I beg you to answer... Let's suppose that I'm asked to find the fourier series of some kind of function, over a certain segment $\left[a,b\right]$. Let's say that I'm not interested in studying the properties of the function; instead, I'm required to solve the integral: $$\tilde b_n = \frac{1}{\sqrt{b-a}}\int_a^b dx \, f(x)e^{-i \frac{2n\pi}{b-a} x} \; \lt \infty \quad \vert \quad \forall \; n \in \mathbb{Z}$$ whose results are the coefficients for the function's fourier series: $$f(x) = \sum_{n=-\infty}^{+\infty} \tilde b_n \frac{e^{i \frac{2n\pi}{b-a} x}}{\sqrt{b-a}}$$ Why doesn't the simple existence of the fourier series coefficients imply the pointwise convergence of the series itself to the given function in the given segment?