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Until such features are available in the client, you can find some of the stuff you're looking for at:

http://dominion.lauxnet.com/leaderboard/

On turn 14 (when you played 3 Charms) you bought the Province before the Duchy. Charm's extra card gaining ability only works "the next time you buy a card this turn", which in this case was when you bought the Province. To get an extra $5 card you needed to buy the Duchy first.

There's no turn where you buy a Charm with 2 Charms in play. On turn 13 you buy a Market with 2 Charms in play and gain a Charm and a Count. When I undo back to that decision point, after buying the Market I can gain any two $5 cards which are not Market, which is correct (I could gain 2 Charms).

The cards that came with the physical expansions were something for the Goko version of Dominion Online (two companies ago). And all they did was grant access to a small set of cards (3-5 maybe?), not any full expansions. They were just a sort of promo for the online version.

Click "New Table" at top, "Create Table", "Add Bot" for as many bots players you want, then click "Advanced Options" and set "Allow Spectators" to No. Click "Ready" to start game.

There is an unofficial website with this info:

http://dominion.lauxnet.com/scavenger/?user=Mike%20Thicke

Does resigning affect your rating more than a standard lose?

No, they are the same. From the original rating announcement:
"- The only thing that matters is win/tie/lose (point margin is irrelevant; resigning is the same as losing)."

So I was thinking that if you exit a game (either because you closed the browser or you had a technical problem), and log back in within the 5(?) minute window, you will always get prompted to continue the current game, correct? Is it possible to decline and instead start a new game? The solution is that it shouldn't be. You should actually be immediately put back into the current game. Only after the time window is finished should you be able to start a new game. This is totally fair, because if you were thrown out through no fault of your own, you would want to continue that game. If you did it just to make your opponent wait, you have to wait too. Seems like a no-brainer. (Maybe it already works this way?)

It already works this way.

If you've added each other to your "Friend List", go to "New Table" -> "Create Table" -> "Advanced Options" and change the 'Allow spectators' option to 'Friends only'. Also make sure to increase 'Max players' to 3.

If they go to their "Matching" tab they should be able to join your table, and people who aren't on your Friend List won't be able to.

There's no way currently to be more specific in the restrictions, i.e. you can't choose two specific players, you can just limit joining to all your Friends.

If you go to the "Familiar Cards" tab after logging in and choose "All Expansions" you should see cards from all the sets show up in games with the Bots.

If that doesn't work, I'm not sure what the issue might be.

Can this not lead to abuse?  Playing an unscrupulous player where you request an undo all they have to do is ignore it and wait and they'll win?  Or am I missing something?

You can cancel your own undo request.

*Arena continues asking you to discard even after the last VP have been removed.

You are allowed to keep using the Arena ability even once the the VP are gone, and there are a few scenarios where you'd want to do this (Haunted Woods, planning to return to Action phase with Villa and play a Library (or similar card), likely some others).

*Why does Scheme insist on asking you to top-deck cards after the game is over?

The scenario here is pretty obscure, but with Scheme it is possible to both gain a Madman and keep the Hermit if you choose to Scheme a Hermit and buy no cards. This can impact the scoring of a few Victory cards and Landmarks. I can't think of another case but there could be one.

You have to use the same # of players as the original game, so in this case (2 players) you could replay it with one bot + yourself.

It's a UI bug, if you're lucky enough to have a copy of the card in your hand (or in play I think? not sure on this one), you can click on it there to name it.

It's just a display error, I've checked both of the cases you mentioned.

Once you've subscribed, anyone can join your games and play with the expansions you've subscribed to.