Can Time Machine remove some files completely, and/or sync only to within 6 months? I wonder for Time Machine, if I delete 3 files on the hard drive that I will for sure never want to use again (for example, that are 1GB each), can I tell Time Machine that please don't archive these files and remove them from the history so they don't need to occupy any space?

Can Time Machine also be configure to keep history for up to 6 months (or 4 weeks)?  Or otherwise, how does Time Machine handle it when the backup hard drive is full?  (will it automatically delete the oldest history?)  thanks.
You can add files and folders to the exclusion list - this only prevents future backups from copying those files. You can script these exclusion lists with terminal in Lion using tmuil.

You can delete all versions of files and folders using the Time Machine interface. This is a somewhat manual practice.

Apple doesn't support (there's no built in setting) to set an arbitrary time limit for backups. Once the drive fills, TimeMachine will delete enough backups to start the current backup and then warn you that files have been deleted. It will do the deletion silently if you don't ask to be warned in the preference pane. You have to monitor the drive if you can't afford Time Machine automatically deleting files when it needs more room.