Does it work with mod X ?

It should. There are very few mods with which Fluid Permutations cannot work and in some of those cases compatibility can be added. The only concrete exception are mods that have a building with a fixed recipe - Fluid Permutations cannot change it, since it is fixed.

Can I add this to my existing game?

Yes!

Whenever I try to rotate a recipe added/modified by another mod, the recipe's ingredients/products change.

This is most likely due to mod loading order - the other mod loads after Fluid permutations and FP generates permutations before the other mod modifies the recipe. Report the issue in the discussion tab. Please specify what the other mod is or provide the list/chain of mods that have modified the recipe. Settings -> Interface Settings -> Tooltips -> Show mod owners in tooltips.

Factorio crashes/errors

If the crash/error is related to Fluid Permutations please report the problem in the discussion tab and provide information about it. What were you doing when it happened? Provide the error message - the text from it (either copy it or make a screenshot/snip). If the game crashes provide the messages from the current log file.

Is this different than mod X ?

Probably. There are some mods that are similar, but they do somewhat different things, work in a different way or are not doing some of things that I want from such a mod.