You can't do bounties, or at least, can't do lots of bounties. Also pirates will send trade fleets, and sometimes those trade fleets will fight other faction trade fleets and cause shortages, which is annoying, but in general, there aren't any serious downside right now. I think that could change in the future though.
I'm pretty sure the LP cells don't have any complicated supply mechanics, they just pop up if there's too much stuff (AI cores, technological industries etc.) in your colony, and they are associated with some LP base that you can destroy to knock them out for a while. Also, I don't think there's really any significance to the LP because all their terrorist cell events are bugged to always fail right now, so you can just ignore them completely.