Knowing the inner mechanics is definitely helpful, though with the high level of randomization, new and unexpected combinations still make it a non-trivial affair!
Yes, everything is independent. If you have, say, two of the same proc that procs with 50% chance on hit, on a hit you’ll roll for both of those and both can potentially proc at the same time.
I realized a bit too late in development that having so many abilities able to proc with the same name on the same turn meant that the combat log would be difficult for the player to parse. This means that the player potentially misses out on information crucial to learning the game’s mechanics.