Yes! Worst that can happen is they say no. I'd include a piece of fan art based on one of their books to show them what you can do.

I'm getting a lot of this in my DMs at the moment. I don't mind contact, but one or two are getting quite pushy about it. I say I will look for art when I need it (too many things going on right now). However. If someone sends images with price guide, that is super helpful. I need to know whether I can afford it before I will negotiate the details, if you know what I mean. A website, with details. Samples, prices, time scales and some details even about the process is a big yes for me.

Depends what you mean by collaboration. If they have artists they regularly work with, they are probably not looking for new ones, particularly if they write series and therefore would probably want to stick with the existing cover artist. If you want them to write a new story purely for you to show off your illustrative skills then no, they are unlikely to be interested unless you are already well-known yourself.

Would usually go through their agent - it depends how popular they are

The main problem is you would have to convince him/her of several things which will be difficult: 1) That an illustrated version of an existing book would open up new markets; 2) That a book they're writing now would be better with illustrations when they've been selling okay without; 3) That the money they pay you will bring in enough money beyond the expense.

I suggest not putting the author on the spot. You could do art based on the book and put it on your social and tag them and mention their book (many people do this with SJM though that’s mostly do boost their own algo). If the author likes it, they’ll say so. Then it would seem less pushy to suggest a collab.

As an author myself the best thing to do to get my attention would be to send me some fan art. If it’s really good then the author will reach out.