Choices of $q$ and $f$ for RLWE-based constructions
Why use $f(x) = x^{2^n} + 1$?

This is a very special polynomial called a "cyclotomic" polynomial. Cyclotomic polynomials are very interesting, and I recommend reading up on them separately  (wikipedia). One very useful property is that the $m^{th}$ cyclotomic polynomial has roots that are all the $m^{th}$ primitive roots of unity. For any $n \geq 1$, the polynomial $f(x) = x^{2^n} + 1$ is the $(2^{n+1})^{th}$ cyclotomic. Power-of-two cyclotomics are even more convenient because they allow for much faster NTT algorithms (see Cooley-Turkey style optimizations, where having a power-of-two length is important for the recursive structure of the algorithm). Here is a nice paper on NTT optimizations.

Where does CRT come in?

This will also answer your third question, since $q$ need not be prime, and this is necessary to implement CRT optimizations. The problem that arises with RLWE-based secure computations is that the modulus $q$ must grow to be much larger than the typical machine word, which is usually 64 bits. Since $q$ need not be prime, we can use a modulus $q$ that is the product of many small primes $q = \prod_{i=0}^{\ell-1} q_i$. CRT makes use of the following isomorphism:
$$ \mathbb{Z}_q \cong \mathbb{Z}_{q_0} \otimes \mathbb{Z}_{q_1} \otimes \ldots \otimes \mathbb{Z}_{q_{\ell-1}} $$
This is what allows the implementations to work with each $q_i$ separately instead of the full $q$. This also allows for parallelization. For more information on this isomorphism and how it's useful in RLWE implementation, I recommend this paper.