"If you have Google Labs enabled, you can try out the advanced IMAP controls feature. Among other things, it allows you to define which labels/folders you want to sync.
This might help you with Question One.
Note also, that part of the reason that this is confusing is that Google designed Gmail from the ground up and it's built around labels -- each message can zero, one or many. IMAP is built on hierarchical folder -- each message is in one.
When Google added IMAP support to Gmail, they had to find a way to graft labels onto IMAP's folder protocol. This article in Gmail help has a nice table showing how IMAP actions map to actions you would perform in the Gmail web interface."
See some more details on Gmail settings at Greyed out emails in Mail.app. (Be sure to understand how the Gmail Trash works, and how to set up its SMTP server.)
