Is there a limit to the number of requests I can make with the API? I see there are limits to uploading and the total number of videos (2000) you can have, but is there a limit to requests I can make to, for example, the search query methods?
http://code.google.com/apis/youtube/2.0/reference.html#Searching_for_videos
YouTube APIs FAQ: Are there any limits on the number of operations I can perform using the API?
Yes, there is a quota system in place to prevent abuse of the API.
Check out this discussion. Also, you'd want to look at the developer's guide pages regarding quota errors and caching data to avoid quota errors. As you can see there, you will also have a limitation in the number of calls from the same caller in a short amount of time.