I tried to get some extra trade in one game by gifting a frighters to an AI after all of my slots were filled. The ship was adjacent to one of my planets so it would seem obvious that they would have the route with me, a bit of a gamble i guess. However the ship never moved.
Other times I might try to get an AI to take out another's resource starbase by giving it a ship next to it while they are at war. Sometimes the just fly by it. Perhaps they look at the strength of the starbase. Generally though i find the AI uses the ships i sell or give them. Mostly i lose track of them.
The freighter issue is fairly obvious- since it was made on one of your planets, that's the one the trade route would start from. The AI probably can't figure out where the trade route would start from once you gifted it, so the ship wound up just sitting there, useless.
As for the resource starbase, that AI may have figured out that you were planning on swooping in to claim the resource the instant the base was taken out. They'll usually not go after mining starbases at all unless they have some constructors ready to send to the available resource, so I'd park a few constructors with a ton of engines outside their sensor range. That way, they think they can claim the resource, take out the starbase, and I can swoop in from the mists to plant my flag first.
As for the Terran Ranger at the top, they probably retrofitted it or disbanded it. Lucky ships tend to be absolutely horrible from design standpoints, so I treat them like free upgrade-ready hulls.