Even though the aircraft appears to be abandoned, I would start with the legal aspects before sinking any $$$.
Yep. The lay term 'theft by finding' springs to mind.
I note the OP called it 'abandoned' and I suppose it's hard to lose or misplace an airframe, but at what stage does it get to the stage that someone else can claim ownership? If not the original owner, does an insurance company have some interest in it?
The late Tom Kruse OBE (the Birdsville mailman) got mightily aggrieved when he found out that the Copley Crew had retrieved his Leyland Badger truck out of the desert (albeit with the intention of restoring the historic truck.) Tom had 'parked' it there (on someone else's land) some 20 years before and it was subsequently deteriorated but he still considered it 'his!'