couple things:
(1) no, you can't get .Mac to use your filters
(2) your duplication problems might be solved by tweaking the settings you have for "remove messages from server" on both .Mac and at home.
(3) the way I do it, I just leave Mail.app running on my home computer, and add a rule after all the other rules, to transfer incoming mail to my .Mac account's inbox - that way I get the benefit of the junk mail filter, even when I'm checking from .Mac - but it does require my home computer to be on and running Mail all day.