Question one: Yes. Which is why it's easier to EV train after the Elite Four, so you don't have to go through a specific place and battle trainers.
As for random 1-32: No. That 1-32 is actually 0-31 IVs, and is something completely different. If you were to catch two Geodude at the same level, they might have different stats due to IVs. That's a pretty rough explanation, Serebii's is better.
Question two: No. There's a minimum a Pokémon can have. If you were to catch, say, a Geodude at level 99, it would have specific stats. If you used a Rare Candy (giving 0 EVs), it would level up and gain stats (let's say we didn't evolve it). However, it would have gained many more if you were EV training, instead of using the Rare Candy.
Finally: No. Pokémon caught in the wild have not gained any EVs. It makes no sense, but it's true. The reason its stats are higher is what I mentioned answering your second question.