This sounds like a great idea. But I share the concern of others here about being TOO free with when the card could be cashed in. If you can cash them in during an opponent's turn, you could run into all sorts of timing issues that would cause a rules nightmare. Say an opponent has just played Tribute. You know you've got a good card on top of your deck somehow, so you cash in your Duration+ card that gives you +1 Card. Does it take effect *during* the resolution of the Tribute, or must it wait until the Tribute is finished resolving? I'm sure you could come up with a rule that would cover this, but my point is that, whatever rules you create, there are probably always going to be a bunch of corner cases (including possibly with future official cards) you haven't thought of, where it really makes a difference.
Beyond that, there are practical considerations. Say an opponent is zipping along with his turn. He plays one card, then announces another card before you have a chance to butt in and say, "Iplaymyduration+cardnow!" Do you get to play your card before the effect of his second (in which case you benefit from unfair knowledge of what he can do next [or MUST do next, even if your duration+ play changes his mind), or must you wait until after (in which case a strange race element is introduced to the game)?
Reaction cards allow you to play during an opponent's turn, but the timing of when you're allowed to play and in what order everything resolves is strictly regimented.
So let's say you can only cash in your Duration+ cards on your own turn. I still think there are going to be problems if you can do so WITHIN the resolution of any other card (such as Throne Room, in your double Tactician example, cool as that may be). Imagine cashing in a +1 Card bonus in the middle of a Golem play: you set aside a few cards, get down to the last one in your deck, which you happen to know to be your only Gold, cash in the +1 Card, and then resume the Golem's search for action cards. You could do similarly weird things if you interrupted Navigator, Lookout, Warehouse, Cellar, Vault, Masquerade, and numerous other cards in the middle of play. Although I'm struggling to think of an example that's genuinely broken, again my point is just that the interactions are so potentially wild and unpredictable that I'd be afraid I wouldn't be thinking of some broken interaction. Of course, there's no way to know anyway, without knowing what the specific cash-in effect on a specific Duration+ card is.
I'm a lot less nervous if you can only cash-in during your turn, and between the resolution of any given card. But now let's move on to your Ultra-Haven example. Does setting aside an Action allow you to play that action card even during your Buy Phase? What if that card is Golem? Now you're forced to play additional actions during your Buy Phase, too. What if you've already bought a card, which means you're not allowed to play any additional treasures during your Buy Phase, and Golem turns up an Adventurer? Now you have to dig for treasures and put them in your hand, but you're unable to play them. Yet again, I'm struggling to think of a genuinely broken and/or ambiguous rules issue, but it sure is confusing -- you'd have to know the rules extremely well to figure out how to play situations that might commonly arise.
Could you play an action card set aside by Ultra-Haven during your Clean-Up phase? Definitely all sorts of strangeness there. Say you play an Outpost. You draw three cards for your next hand during your Clean-Up phase, then cash-in your Ultra-Havened Golem, which turns up a couple of Treasuries. You get to draw two additional cards and have $2 to spend in your Buy Phase of...which turn, exactly? The current one, whose Buy Phase is already over, or the next one?
To make a short story long, I really think you need to restrict Ultra-Haven to action cards only, which can be cashed-in only during your Action Phase, and not while another card is resolving. You could make it so that an action is required to play an Ultra-Havened action, or you could make the cashing-in free. I think you can still manage double Tacticians this way, though perhaps not as easily: play a Tactician from your hand, then cash-in your Ultra-Havened Laboratory, which draws a Tactician and another card. More easily, if you have two Ultra-Havens, play a Tactician from your hand, cash-in any drawing action you'd previously Ultra-Havened, then cash-in an Ultra-Havened Tactician.
With those restrictions, I think you've got a terrific idea. You need restrictions on play options anyway. Strategy and interest comes from the restrictions, not so much the power. Conspirator is an awesome card, for example, but would be ever so much less interesting if it cost $5 and ALWAYS offered +1 Card, +1 Action. And how much more boring would Grand Market be if it cost $7, but you could use Coppers to buy it?
Even reined in, Ultra-Haven would be a powerful card. It's almost more similar to a Native Village rather than a Haven. Like Haven, you get to select the card you can set aside and you can set aside only one. But then it becomes like a Native Village, where you can cash-in whenever you want, at the best opportunity. It'd be pretty great at pairing up Treasure Maps, or Tournaments with Provinces, or Barons with Estates.
Going back to the vanilla Duration+ idea, imagine how powerful a simple +1 Buy would be if you could choose when you cashed it in? +Buys go unused a lot, just because you need to overbuy +Buy a little just to ensure it'll be there when you need it. But if you could choose when you got your +Buy, you could devote more turns to picking up Conspirators and fewer to picking up Markets. And you wouldn't have to worry about drawing your Bank without the ability to use all the money it gives you.
It's definitely a great idea. I just think some of the interesting possibilities need to be sacrificed for the sake of keeping the game sane and playable.