Evasion is, as the question itself notes, a poor example—there is no need to have evasion in order for improved evasion to function or make sense, and the rules don’t indicate any requirement between them.
Mutagens and alchemist discoveries are a far better example. And the answer there is... that Paizo can’t seem to make up their minds. As written, they work—discoveries check if you have something called “mutagen,” and you do, and then improve something called “mutagen,” which is the ability you are using. Nothing specifies where the “mutagen” had to be coming from. But the FAQ seems to—usually anyway—come down against such synergizing across classes. The FAQ being the confusing mess that it is, this isn’t clear—it’s a hodgepodge of seemingly-specific questions getting vaguely-defined broadly-worded answers that pretend the rulings already exist in the rules, which they do not.
Which ultimately means this is something you have to ask your GM, and they should handle questions like this on an ad hoc basis. Most of the time, it should be fine, but sometimes it won’t be. For your examples here, say, I have a hard time seeing why any GM would prevent it from working. The FAQ tends to err on the side of “caution” by leaning towards “no” for most questions like this, but that is done primarily because it is the “safe” option from a publisher’s standpoint. They don’t have the time, interest, or inclination to thoroughly vet combos, so rather than risk endorsing something game-breaking, they just say “no.” A good GM can, and should if they have the time, do better than that.
