Cool Reader allows to remap the buttons. I actually have next and previous buttons swapped.
You can assign various actions to the buttons for single press, for double press and for long press. So 3 actions for each button.
But please test if the buttons also work in the file manager and in settings.
Go to the list of your books and try to go through it page-by-page. Or in settings.
For example, if you want to swap the next and previous buttons you have to: - go to settings
- then to the tab with the picture of the hand with the finger
- then to "Key actions"
- then to, let's say, "Prev button"
and then you have a long list of actions to assign to it. Try navigating through them with next / prev buttons. Do they work there?
You don't have to go so deep, you can probably test the buttons in settings at the top level - certainly in landscape, not sure about portrait.
As to selecting Cool Reader as the default reader.
I use a different approach. I just launch Cool Reader and never leave it

So I select a book while already in Cool Reader and not what reader to open for a book.
I often use "Recent books" list. A long tap on a book in that list opens a list of options including one to go directly to the folder of the book. I find it very useful for opening new books: if they are in the same series and thus in the same folder, or just in folders close to navigate to.
My main concern is this: how to switch off the indexing of the books by the native software (because it takes a long time and cannot even handle too many books). If you use Cool Reader this way - you don't need anything else to handle the library. On linux devices, using hidden folder usually works fine: if I make the root of my library folder structure a hidden folder, no indexing is performed.
So I have all my books in a folder
.BOOKZ
The dot solves the problem.
But on the android reader T2 a different hack was implemented, requiring a special flag file to be present in the book root directory.
However, the initial dot could work with android as well because the dot also makes the folder hidden.
Could you test this: put some books into such a hidden folder and see if those books are detected and included into your book collection (or whatever it is).
They should not be detected, but still accessible from Cool Reader - that would be perfect.
And if you could test yet another thing: is the external memory card accessible from Cool Reader?
I am asking this because some time ago there was an issue reported with the official Cool Reader (not from jotas) - the external card was not accessible from android 4.x (or from some more specific 4.x.x version) but everything was OK with earlier android versions.
I don't know if it was fixed and if Cool Reader by jotas was affected at all.
Perhaps it was an issue with android itself.
Perhaps the external card needs to be properly mounted first or something.
It works fine with my android 2.1 T2