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)

Relevant cards

Note: All applicable cards may not be listed below.
SM5_134