1) What purpose do multiple races have in a rpg? Or alternatively, what should multiple races add to a game if they're done well?
They serve two purposes in my games. The first is to be a base identity to be something "other" than human, to explore a role, or an idea in the game. This is also served by Aliens in an SF game. The second purpose is to be a source of conflict, This conflict can be different cultural goals at cross purposes, or irredeemable, savage, servants of evil (guilt free targets), and that depends on the background and type of campaign I am running.
2) Do you prefer multiple races at all in your rpgs?
Depends on the campaign, but in most a couple/ three is fine. Too many and it feels like a "reskinned D&D 5e campaign, and "humans only" feels like a well known Fantasy TV series, which works in it's own way, especially since in both cases I tend to run things fairly low fantasy Heroic level campaigns.
3) What fantasy race pet peeves do you have? Why?
The Tolkien Trifecta. it has become the "Default" for fantasy since the three little brown books were published by TSR in 1974. Now on the positive side. D&D and Pathfinder have broadened the selection and types of creatures people can play as Player Characters, but the Tolkien Trifecta is still paramount in those systems. Talislanta had no elves, but looking though those books, they had some that fit the bill, somewhat. Can't someone come up with something new, or at least play with them a little. I will admit to having games where the Tolkien Trifecta were present, but those were mostly "out of the book" adventures.
Another Pet Peeve I have is playing the Non-human Race as a Human stereotype, or a Human in a funny suit. Examples would be the Hippie Elves, The Dwarvish "Soccer Hooligan", or the Hobbit Mafia B&E specialist. I do recognize that not everyone can role play, and still be an asset to the game, but I do expect folks to do a bit of reading about the campaign background and how those cultures worked. I also don't like it when the other "races" are played as joke characters. (messes up the grimdark of the campaign...Kidding just a little bit).
4) How many is too many? Too few?
Between 3 and 5. Fewer than that begs the question of why hasn't one or the other just taken over and made the other extinct. More than 5 (with some exceptions), and the game feels like a costume party, as none of the races will acquire enough development for them to feel like a different culture and biology.
5) What do races represent in you games if anything?
In my game? Depends on the race. The Jaggiri were a source of conflict as they were not defeatable one on one, and were a "Boss fight" against a party. Culturally they were supremacists, and looked at other races as either threats, or economic resources for their territorial expansion. But they could be out thought and out negotiated, and were amenable to diplomacy. They started in the game as a a demonic horror released upon an innocent land. And ended the game as slightly annoying, but decent to know. The Lupines are a stand in for any more primitive culture, except with some tweaks in their physical capabilities and a few broad general mental frameworks that make them, while comprehensible to Humans, different enough that their motivations often diverge from most humans. They do this without having to get into disagreeable situations with the current youth about "cultural appropriation". I roll my own races in games, and after a long history of reading Science Fiction, colonial era adventure stories, and History, the different races basically boil down to creatures from a specific or somewhat general environment that want to eat, breed, raise young, and pass on their culture to the next generation. I had a conversation with Animation Legend Ray Harryhausen once, at SDCC, and he told me, that a Monster was just an animal in conflict. What does it do when it's not angry, and how does it live? Think about that and you then have a more fully fleshed out monster. They also serve to scratch my world building itch, as they will have place names and settlements that are all their own. So Fantasy, or Alien. They signify that this place is "not of this earth", even if I swipe large sections of history.