Interesting set of cards....I just have a few off-the-cuff comments based on initial impressions (playtesting might yield different results):
Penal Colony
I'm not sure making Estates worthless really adds much to the game other than providing a slight disincentive for rushes. I think Metropole adds a bit more depth than this.
Grand Duchy
Interesting idea but I'm not sure it will play out as intended. It's almost like going from 8 Provinces to 16 in a 2-player game, except it takes Duchy-dancing to a different level.
I think you would take this over Enclave if they are on the same board, which could be problematic depending on your point of view.
Sabbatical
I'd be curious to see how this plays out. It sounds useful if you know your opponent has a good turn coming up and is threatening to win, but given that they could buy Sabbatical on their turn, I'm wondering if this would just end up delaying the inevitable in most cases.
Hospital
It could make for some interesting interactions even if there are no debt-cost cards. Buying Death Cart, Ritual, or Vineyards, or emptying out the Curse pile become a lot more attractive.
Ekklesia
This is overpowered. Even if all it did were Exile, it's already much better than Cathedral. The option to gain a card to hand from your Exile mat makes it even better.
College
This is useful but it seems overpriced relative to other $5-cost projects. If you can get to the point where you are drawing your whole deck anyway, it's much less helpful.
Extension
I'm curious as to why you prefer to make it vulnerable to hand size attacks rather than draw at the start of your turn.
Letters Patent / Tax Collector
I'm also curious how this would play out. You can't really afford to let your opponent keep Tax Collector for several turns, but since it only triggers at the start of their turn you could probably afford to let them have it for one turn. If it ends up going back and forth between players throughout the game, then it's effectively like players had their -1 Coin token each turn. I'm not sure it would make the game more fun or interesting.
I imagine it would be less enticing in multiplayer games.
Way of the Mongrel
This is definitely more powerful than the official Ways. You essentially have the choice between playing your $5+ cost cards normally or as a Band of Misfits. It does encourage having higher cost Action cards in your deck, whereas with other Ways it's often good to spam cheap Actions. What is the reason for the "non-Duration" limitation?
All your Base
Seems like games with this and Mountebank (or even other cursers) would be pretty fast. One potential concern could be games with Pirate Ship and no other Action-based economy, where there could be scenarios where you wreck your opponents' economy and there aren't enough Treasures in the Supply. I don't think it's likely to occur in a 2-player game, but could happen in multiplayer games.
How would this work with Run on the Bank?
Run on the Bank
Seems like an interesting idea, but my main criticism would be that it makes several existing cards and some Events useless (e.g. Bandit, Soothsayer, Explorer, Enclave, Reap, etc)
Does Gold count as an empty pile? I think a rule clarification would be needed.