>This is not in the Supply.
Good; this protects against Ambassadoring them away.
But you could still Masq it.
That seems OK to me -- having one card that creates an unusual interaction is fine, and Masquerade already does that with other cards.
Right, I figured that the Masquerade interaction wouldn't be a deal breaker, espeicially since repeated plays of the Masquerade may cause Cave-Ins to come back to you. Unlike a Curse that you get passed, you can't immediatley trash the Cave-In.
Island, and to a lesser extent Native Village, can also get Cave-Ins out of your deck. There's also Trader, which allows you to forego Cave-In for a Silver.
Island I'm OK with. I didn't even think of the Native Village interaction, but I'm fine with that one, too.
Trader is the only one that gave me pause, and I had considered having using "move a Cave-In to your discard pile" rather than "gain a Cave-In". For the OP, I decided to post the simpler gain wording and let the Trader interaction ride for now. Just as Watchtower can do some crazy things with several Dark Ages cards, I could live with Trader being crazy good with these cards. If in playtesting it turns out not to be fun, I'll revisit it.
Idea seems good; I like it. Cave-In itself is a bit bland.
Spelunker looks OK. I think it is similar to Scavenger and Sage. More immediate than Scavenger (not to mention action instead of coin), more control than Sage. $3 seems alright, especially because there is a tension between balancing number of Spelunkers and number of Cave-Ins.
Mining Team is a bit tricky. It is REALLY powerful draw. If you reveal a Cave-In, it is still a Warehouse, which costs $3 on its own. Still not that bad (especially because you have at least one dead card to discard -- that Cave-In you drew). Then again, having that Cave-In in your deck at all is a penalty. $4 might be OK.
Diamond Deposit feels too swingy. Might still be OK though? I have this feeling that it could probably be abused in certain combos, but I'm not sure.
Fortify is another super swingy card. No negative side to using it, really, since you can usually replace the card you trashed. But say one person hits a $5 to get a Province while you hit a Cave-In and don't even get to remove it from the top of your deck. 
Architect is probably OK as well…
Thanks for the detailed feedback! I'm glad you like most of the cards. As for your criticisms, I agree that Diamond Deposit is swingy, but I'm guessing it'll be Counting House-style swingy, which I'm fine with. Fortify almost definitely needs work, though. I mean, it might turn out to be fine, but you're almost certainly right about it being way too swingy. Originally, I was going to have it be, "Trash the top 2 cards of your deck. For each one, gain a card costing up to $1 more than it." But that has its own set of issues. I'll work on other ideas for a Remodel variant. Thanks!
With the on-trash mechanics, you could be more direct with this too. rinkworks' article mentions that a popular solution to Blood Money is to make the cursed card untrashable. The main complaint about that is that "this cannot be trashed" does not seem to jive well with existing Dominion rules. Fortress is an example of how it can be phrased to fit. So you could make a powerful card that comes with a VP penalty and is "untrashable" in the same way as Fortress.
Or you could make it trashable, but have the on-trash be a penalty.
"When you trash this, gain a Curse."
"when you trash this, gain 3 Coppers."
"When you trash this, each other player may gain a card costing up to X."
Yeah, these could all definitely work. But once I struck upon the idea of Cave-In and the cool ways cards could interact with it, I was hooked! These other ideas are all very valid, though.
Finally, I had one additional, very simple card idea.
Boomtown
Types: Action
Cost: $4
+2 Cards. +2 Actions.
-------------------------------------------------
When you gain this, gain a Cave-In from the Cave-In pile.
As far as drawing useful cards, this
should be more or less equivalent to a vanilla Village. Every shuffle, it draws you one more card than Village would, but you also have one more garbage card to get through each shuffle, so it evens out. In reality, there are lots of cards that benefit from a large handsize, even if several of the cards are useless. Hence, it costs $4 instead of $3.