Can I choose to play Unidentified Fossil as one of my starting Pokemon?
No, you cannot. It's considered a Trainer while in your hand, so it doesn't count as a Basic Pokemon during setup.
Source: TPCi Rules Team (2018-02-08)
Can I use an Ability to move enough damage counters onto my Unidentified Fossil to knock it out, but then discard it from play before it gets KO'd?
No, that won't work. As soon as you're done using the ability you would check for KO before being able to discard the Unidentified Fossil.
Source: TPCi Rules Team (2018-02-08)
If an opponent KO's my Unidentified Fossil do they get to take a prize?
Yes, that's part of being treated as if it were a "Basic Pokemon". Your opponent takes a prize when Unidentified Fossil is knocked out, but not if you discard it from play.
Source: TPCi Rules Team (2018-02-08)