Hello, does the Junk Droid controls an objective if it remains in play after all Ugnaughts are defeated and it is adjacent to the terminal/objective?

Yes, it counts for control but cannot activate.

Does the Junk droid activate if an eUgnaught uses SPOT WELD? or its just enters play and is activated at the start of the NEXT ugnaught activation?

All deployment cards enter the game not tapped. This includes the Junk Droid.