The default assumption seems to be that characters don't know much in-character about other classes' abilities, or even their own much in the future, or sometimes even in the present, and that they don't have an in-game concept of class.
That said, it would not be unreasonable to expect that a character that has undergone formal training(*) would have been told a lot of this stuff, especially about their own class. This also depends upon other aspects of the character's background, and is probably region-dependent and organization-dependent as well. For instance, Cheliax and similar competent dystopias probably have all this stuff catalogued (although how much and how truthfully they instruct you in it is another matter), and similarly, scholars in some world hub like Absalom probably have a pretty good understanding of this stuff, having figured out the mechanics by a long history of trial and error (possibly even to the point of using polyhedral dice for mathematical modeling purposes :-) -- after all, on Earth we managed to develop the IQ concept for measuring Intelligence, even though this seems to have fallen out of favor since the 1970s), and in the case of dystopias like Cheliax and some of its neighbors, perhaps even adjusting their training programs to hyperoptimize their minions (even boosting abilities seen as necessary to the point of damaging abilities seen as unnecessary and/or troublesome -- Mengkare/Hermea is another candidate for making a fine art of this, as part of the Glorious Endeavor eugenics program). Of course, having this training and figuring out where somebody else fits in to it would be different matters. If you want to tie it to relevant skills, Knowledge (Arcana), Knowledge (Religion) and Spellcraft seem appropriate for figuring out spellcasters, while Profession (Soldier) seems appropriate for figuring out martial types (and that skill is currently a bit short on common uses, so this seems like a good addition). More average people in more average places probably fit more with the default assumption (at least until they get enough ranks in the relevant skills); training like that above should probably be reflected in ranks and possibly Skill Focus in one or more of these skills, and possibly relevant skill-boosting traits.
(*)Arcane Academy, Bardic College, Church/Temple, Military Academy, or Special Operations Training, for example.
There is not a hard rules answer for this. It's going to be left up to GM fiat.
Personally I allow characters general knowledge about their own classes, but I do not allow them general knowledge about other classes (without a knowledge check at least).
I will go with Campin here , since i dont remember any hard rules for this.
Usually when i play a PC knows his class really well (for example a caster would know the future spells of the class and thus have no problems selecting scrolls...) and has a general idea about others , like the PC would know paladins are great against evil foes and have strong on "saves" in general , unless the GM says a class is really rare...
It depends. As has been said, there's no rule, just reasonable interpretations.
Is it reasonable that someone who apprentices and learns 1st-level wizard tricks knows some day fireball might happen? Yes. Because "everyone" will have heard of wizards casting it.
But esoteric abilities or optional 20th-level abilities? Less clear.
I'd use various Knowledge checks to determine if the PC has heard of precedents.
Keep in mind very few people on Golarion have ever made it to level 20. So I agree with Anguish on this one.
The more "formal" the training required in a class the more likely high level abilities are to be known.
Wizard abilities will be more well known than Sorceror abilities for example. This makes it very setting dependent as some worlds have formal bardic colleges, and others very little in the way of structure with master/apprentice or 'learn on the job' approach.