What happens if I reach my daily reputation cap? Does it prevent me from asking questions, or what happens?
No, it just means you won't gain any reputation from upvotes cast on those answers.
You can still post answers, people can still vote on them, and the vote tallies displayed for those answers will still accurately reflect the votes that are cast.
The trick lies simply in the fact that the vote count will change without your reputation score changing.
Also note that answer accepts (+15) are not affected by the rep cap—you will still get those points, so there's still some benefit to answering questions.