When I was offered a class it was 3 to 4 weeks in advance. Although if they have someone drop out of a class they might ask you if it's ok if it's less than 2 weeks.

You will know what certificate and aircraft because they allow you to choose. When they make you an offer they will ask your preference. You can turn down a class offer 3 times before having to re-interview (It was this way as recent as November 2012, don't know if it has changed).