Clocks on devices like the DR-680 and cameras (especially DSLRs) aren't that accurate, so it's likely that they'll drift theoughout the day and there's no way of syncing them (no wordclock/genlock), although the fact it's 2 minutes off seems a bit buggy (is it always by the same amount?). Also have a look on the audio board at dvxuser.com there are quite a few people using DR-680's for production sound
I'll usually try and match the time on the camera and my recorder, but if it's properly slated you should be fine.
You should be ok putting TC onto an audio channel, use a big pad on it (eg 50dB) to prevent bleed. However, it's not standard workflow (especially as you need to buy a plugin for FCP), talk to whoever's doing post first. If post require TC, it's probably best to hire an SD 7 series or similar (and they'll need a lockit/TC slate for camera).