Just some ideas that I'd like some thoughts on.
Nice! My comments:
Auction
This one is just too hard for me to figure out how it would work. But I would suggest simplifying it by having players set aside fewer cards and then use the costs of the cards rather than their value in treasure -- the reason being that the latter requires too many special-case rules to figure out, such as Venture only being worth $1. Bank and Philosopher's Stone? Quarry? You can limit it to non-victory cards to close a possible loophole there, but this way you get to allow action cards, which gets rid of how the card rewards Big Money. Reduce the +$ bonus to half the costs of the set-aside cards (or, alternately, +$ the cost of the auctioned card, which might be better) should rebalance that piece.
But even with these suggested changes, I really can't envision how it would play, and I worry that it might significantly slow the game down.
Storage
Probably my favorite idea in this batch. I like your original version better than the variations you propose in your comments. Is the trashing piece really necessary, though? I'm not sure that it adds anything and might even make the card a touch strong. Without it, the idea is clearer to me: buy a card at a discount but wait a shuffle before you get it. It's interesting gameplay in both the early game and late game. Early, you tend to need those good cards in your deck as fast as you can get them there, but this allows you to get more or better cards if you wait a little longer before getting to use them. In the late game, sure, you can buy a Province, but at the risk you won't get it into your deck before the end of the game. (Interestingly, buying the last Province this way would end the game but without you getting the points for it.)
There's also a risk that when this card comes up again, it'll clash with another important card in your deck or, worse, will be drawn dead. So it's a risky proposition but might sometimes be a really good idea. I also like having another card that allows the playing of treasure during the action phase. That has such great combo possibilities that it's really too bad only Black Market can pull them off. I wouldn't want very many such cards, but another one or two seem like a really good idea.
I might drop the price down to $3, too.
Cursed City
Probably a good card, but I've never tested "gain a Curse; if you do..." variants.
Grim Reaper
If this seems strong, you could limit it to trashing non-victory cards, which will keep players from being able to trash Provinces and Colonies without losing them. Other expensive cards (Platinum, King's Court) aren't a problem, because, while you can trash them for the $ while still keeping them, they do leave your hand, so you can't also play them during that turn. Unless you're able to finagle a way to draw them back into your hand, which I guess is easy if Apprentice is the trasher -- so maybe only allowing the trashed card to be put in your discard pile, rather than on top of your deck, is the way to go here.
Not a fan of the more complicated alternate wording: in fact, I might even want to simplify further and ONLY let the trashed card be discarded. The card is simpler that way while still retaining its gameplay value.
Collaboration
Cool idea, but I wonder if it would play differently enough from Laboratory to be worthwhile in practice. The thing about Laboratory chains is the more you play, the more likely you're going to be able to play them all. Collaboration is essentially exactly this way, except that the difficulty of starting the chain is exaggerated. Once you play that fourth Collaboration, you're indeed drawing more cards than the same number of Labs, but playing four of
either card is almost certainly enough to draw all you need. The one situation where there would be a substantial difference is in a fat, cloggy deck, where lots of extra +cards could be a real help. But in those decks, you're not likely to connect your Collaborations in the first place.
I also see this card playing dramatically differently in 2p vs. 3p or 4p. With 2p, it would be very difficult to prevent your opponent from getting 4+ copies. In 3p or 4p, you can probably make sure no one gains a critical mass of them. This is also true of Laboratory, but Laboratory is plenty useful in singles and doubles. But it takes two Collaborations to equal the power of a single Laboratory. So if players succeed in locking each other out of a critical mass, they've essentially burned a few turns on something that isn't that helpful to them. And if one player manages to get a critical mass anyhow, it's probably game over.
More interesting might be going the other way. The wording won't be as elegant, but a card that offers +3 Cards, +1 Action when only one is in play, and one fewer card for each additional copy in play, might be a pretty interesting card. Or maybe still too powerful, I dunno.
