Pauluk, you got it! When the elctrician came again this summer he didn't fumble around long but took the neutral of the garden hut out of its lug and everything worked perfect since. Each time he tried to reconnect the neutral for determination purposes the GFI tripped immediately.
Afterwards we figured out that the resistance of the long neutral run from panel to transformer was high enough to have some power flowing back in the neutral to the garden house. Up there they must've had a neutral to ground and maybe also a hot to ground fault (the tripping reduced when we threw the breaker). Concerning the freezer I can only guess that this was because it had almost stopped raining when we did this and so when we got the mats installed everything had dried up enough to work.
