Custom kernels for SVM, when to apply them?

I am new to machine learning field and right now trying to get a grasp of how the most common learning algorithms work and understand when to apply each one of them. At the moment I am learning on how Support Vector Machines work and have a question on custom kernel functions.
There is plenty of information on the web on more standard (linear, RBF, polynomial) kernels for SVMs. I, however, would like to understand when it is reasonable to go for a custom kernel function. My questions are:

1) What are other possible kernels for SVMs?
2) In which situation one would apply custom kernels?
3) Can custom kernel substantially improve prediction quality of SVM?

1) What are other possible kernels for SVMs?

There are infinitely many of these, see for example list of ones implemented in pykernels (which is far from being exhaustive)

https://github.com/gmum/pykernels

Linear
Polynomial
RBF
Cosine similarity
Exponential
Laplacian
Rational quadratic
Inverse multiquadratic
Cauchy
T-Student
ANOVA
Additive Chi^2
Chi^2
MinMax
Min/Histogram intersection
Generalized histogram intersection
Spline
Sorensen
Tanimoto
Wavelet
Fourier
Log (CPD)
Power (CPD)

2) In which situation one would apply custom kernels?

Basically in two cases:

"simple" ones give very bad results
data is specific in some sense and so - in order to apply traditional kernels one has to degenerate it. For example if your data is in a graph format, you cannot apply RBF kernel, as graph is not a constant-size vector, thus you need a graph kernel to work with this object without some kind of information-loosing projection. also sometimes you have an insight into data, you know about some underlying structure, which might help classifier. One such example is a periodicity, you know that there is a kind of recuring effect in your data - then it might be worth looking for a specific kernel etc.

3) Can custom kernel substantially improve prediction quality of SVM?

Yes, in particular there always exists a (hypothethical) Bayesian optimal kernel, defined as:

K(x, y) = 1 iff arg max_l P(l|x) == arg max_l P(l|y)

in other words, if one has a true probability P(l|x) of label l being assigned to a point x, then we can create a kernel, which pretty much maps your data points onto one-hot encodings of their most probable labels, thus leading to Bayes optimal classification (as it will obtain Bayes risk).

In practise it is of course impossible to get such kernel, as it means that you already solved your problem. However, it shows that there is a notion of "optimal kernel", and obviously none of the classical ones is not of this type (unless your data comes from veeeery simple distributions). Furthermore, each kernel is a kind of prior over decision functions - closer you get to the actual one with your induced family of functions - the more probable is to get a reasonable classifier with SVM.