This is very likely not going to happen. Donald X. has said that of course variants are find when you play with your cards, but the online versions should stay as close as reasonable to the published game.
The problem is for a computer to decide what an "obvious mistake" is. I trashed at least Provinces before by intent, either to Remodel them into a new Province or to draw cards with it from Apprentice.
What has been discussed, but not implemented are "no new info" undos, where the effect to undo did not give you info you didn't have before. That would have worked if the trashing effect was say Remodel or Forager, but not if it was Apprentice (because you'd seen the drawn cards).
Unfortunately, I don't think there are plans to implement that version either in the forseeable future..
That is an unfortunte side effects of the server restarting (usually twice a day). Your game results have not been lost, they will be taken into account on the next ratings update (around UTC midnight each day).
I assume you are referring to your game vs. Strudelteig on November 7th. In that game you could not activate the bonus portion of Conspirator once (turn 10).
You had a Dungeon in play from last turn, handled that at the start of your turn, then played a Horse, then the Conspirator.
The Conspirator was only the 2nd Action card you had played that turn, so it did not activate its bonus. The Dungeon was in play, but was not played this turn. Doing it's start of turn effect is not considered playing it. That left the Horse and the Conspirator as "played ... Actions this turn".
Thanks for the report. I don't think there has been a new version of dominion.games out for some time. Can you look for and reply with the version of Fiorefox you are using? In the past, sometimes when Chrome was updating they changed things that lead to similar things happening.