"but have all the performers that have been in Avenue Q really study it for more than 3 months or so before they start?
honest question, I really do wonder. "
Well the original cast of the show was made up entirely of people who had made their adult careers out of puppetry, 10+ years of real-life experience each, college degrees, except for Aymee Garcia, but she's from my hometown so I forgive her.

Plus, she puppeteers on PBS's "It's A Big Big World" so she is applying the experience. I am pretty sure she had studied puppetry before Avenue Q though.
As written on this thread earlier, all the cast members are asked to perform a puppet at the audition to see how well they can lip-synch and whether there is hope for them to continue further. Rick Lyon and Jennifer Barnhart have been in charge of teaching the new cast members in auditions/rehersals and even during performances. Just as a dance captain would be in charge of maintaining a show's choreography, the puppet captain is in charge of the upkeep of puppetry and gives notes before shows.
Three months isn't necessarily a lot of time. Think of it as learning a new language. You have acquired some vocabulary and can write and read some sentences after learning for three months, but you can't converse and speak it fluently or write an essay in the new language. (And if you can, please teach me!)
Dean: Can I tell you something?
Lorraine: That depends on what it is.
Dean: I think you're really really pretty.
Lorraine: (after a pause) Ok, you can tell me that.