Does it record to two files if the shows overlap (due to begin/end padding)?
Yes. You can create up to five virtual tuners per tuner card, which will solve this problem.
Does mythweb has a record button on the shedule overview?
This I don't know. I would check, but my Myth box is not connected to the Internet right now.
Does it play DVDs out of the box?
Yes, at least for me it does. (Mythbuntu 10.04)
Does it allow me find© the recorded files to another machine so I can watch them on the road? (searching trough the hashed filenames is no fun)
Sometimes I have copied an episode or a movie straight from the Mythweb interface to some other computer.