Answer to 1:
While Fourier analysis is virtually "solved" in the spaces $L^2([0,2\pi])\quad $, in the sense that thanks to the orthogonality and completeness of the system
$$ \{ e^{2\pi i n \cdot} \}_{n=1}^\infty $$
we can assure that the Fourier series of any function $f\in L^2([0,2\pi])$ converges, namely,
$$ ||f-S_N(f)||_{L^2(\mathbb{R})} \longrightarrow 0,$$
where
$$S_N(f)(x) := \sum_{n=-N}^N \hat{f}(n)\cdot e^{2\pi i n x}. $$
But, while that is true for $L^p([0,2\pi])$ when $p=2$, and for many years was believed to be true for a neighbourhood of $2$, Fefferman showed in 1971 that for every $p\neq2$ there existed a function in $L^p(\mathbb{R})$ such that its Fourier series doesn't converge in $L^p(\mathbb{R})$. This basicaly means that you can't decompose a given function $ f\in L^p([0,2\pi]) $ into simple frequencies via its Fourier series unless $p=2$.
In that concern, a major goal in harmonic analysis is to find a way to decompose any given $f\in L^p([0,2\pi])$ into simple frequencies regardless of $p$.
Moreover, even considering $p=2$, what happens when you swap the set you are considering your functions is, that is, can you swap $[0,2\pi]$ for any other subset of $\mathbb{R}$ and have everything work out nicely? The answer is yes in the case of any compact interval, but no in the case of $\mathbb{R}$ itself.
This leads to the necessity of defining a different method for the non-compact-supported functions $f \in L^2(\mathbb{R})$: the Fourier transform:
$$ \hat{f}(\xi) := \int_{\mathbb{R}} f(x)e^{-2\pi i x \xi}d\xi $$
Giving it a glance one can easily determine that the definition makes sense in $L^1(\mathbb{R})$, but a priori you can't define it the same way for $p\neq1$. So,as you can see, just the fact of defining the Fourier transform in the space $L^p(\mathbb{R})$ is a challenge when $p\neq 1$, let alone determine whether or not the equality
$$ f(x) =? \int_{\mathbb{R}} \hat{f}(\xi)e^{2\pi i x \xi}d\xi $$
holds in the sense
$$ ||f(x) - \int_{\mathbb{|\xi|<R}} \hat{f}(\xi)e^{2\pi i x \xi}d\xi ||_p \longrightarrow 0 \quad \text{when} \:\: R\rightarrow \infty.$$
You can see this as analogous to the series case.
As for question 2:
Fourier analysis interconnects two areas of math hardly related otherwise: analysis and discrete math via the identification
$$ f \rightarrow (\hat{f}(n))_{n=1}^\infty $$
that is, f and the sequence of its Fourier coeficients. In fact, in the case $p=2$ you get a 1 to 1 identification between functions in $L^2([0,2\pi])$ and the sequence space $\ell^2$.
Needless to say, harmonic analysis is very useful when it comes to image and sound processing and compressing algorithims.
