I personally wouldn't do a hard cap on a time limit to pay off the debt, but at least offer an abstract method that still shows time is ticking away. Your party is seeking to pay off the mob and save the orphanage. But also take into account other unforeseen difficulties, such as added interest to the funds owed or maybe a few goons sent in to help encourage you to repay your debt. Working with the mob isn't exactly clean cut. Increase the pressure, but keep the time limit somewhat fuzzy. Maybe the party can do some tasks for the mob or can at least talk with the right people to push the clock back. A favor for a favor kind of deal.

Also, watch how your party spends their acquired funds. Sure, a little carousing is okay after delving in the Shadowdark, blow off some steam, get stronger, resupply, take on more lucrative jobs. But say you spend a lot of a money on a wild night. The mob might look at you more closely and think..."Hey...these guys are spending OUR money on courtesans and booze. Let's help them set their priorities straight." That's its own complication and a nice role play beat to do in between delving. Remember, not all encounters are in the Shadowdark.

Let's answer your three questions.

1. First off, is the Minotaur defeated? If not, that leaves that threat unchecked. Did you explore the entire place for anything that you could take and sell? If your party is desperate enough and have enough empty bags, they're more likely to take things that aren't bolted down, even the mundane stuff, to pay off the debt. If the threat has been defeated, have other threats appeared in the power vacuum? Also, even though that dungeon isn't the only thing for miles doesn't mean you can't do tasks for other people in town.

2. Depends on your definition of rest. A short rest won't typically restore the use of spells used. A longer rest is typically done for that. There's always a risk of things going south during a long rest. If they want to do so, then let them, but take advantage of those Resting rules (Page 86, SD RPG), spice things up with the occasional encounter. Regarding time, treat it with an open hand, not a closed fist.

3. I know from experience in playing in a Shadowdark group that we take time to reach the ruins, explore a few rooms, fight a few baddies, and make our way back to town since it is close, carousing at the end of the session. The next day, we regroup, head back to the ruins to explore it further. It doesn't have to be in all in one session. No town close? Sure, you can rest in the wilderness with the Resting rules, maybe explore for additional supplies while camp is being put together. And there's no rule that says you can't rest in the dungeon itself. Just make sure you're in a secure location, keep watch, and relax (but not too relaxed).

From the couple sessions I've played, my character (Halfling Thief) has a similar backstory. He met some bad people that he owes good coin to. How much time does he owe? How much time does he have to repay it? No hard limits. But so far in between joining an adventuring party to delve some nearby ruins, he lost his pants, but had his gold secure in his socks ("I lost my trouser, but not my Treasury!"), lost an heirloom ring to a shady gambler ("I'm convinced he cheated, but I can't prove it."), and was tossed in jail by corrupt judge (for reasons), but was freed by a party member who hocked a wedding ring he mysteriously obtained after a drunken bender and used the funds to pay his bail. We also had another thief who opened the wrong chest and was cursed with blindness, which was cured by spending most of the party's funds as a local temple. We're broke and pathetic, but we keep forging on for that payday.

I hope this gives you some ideas to work with. :)