No. The game auto-saves during the transition between levels. If you play for twenty minutes, then quit without exiting the level, the next time you resume, you'll have to play start from the beginning of the same level.
On the plus side, levels can often be completed in under ten minutes, so if you're in a hurry and just want to finish your missions so you can exit the level and safely quit the game, it's usually pretty doable, if maybe a bit risky. On the other hand, if you're a "completionist" who likes to get the maximum benefit (XP, money, cool items, etc.) out of a level before moving on, a floor can easily take 30 minutes or more -- particularly if you're trying to play stealthily or expose yourself to as little risk as possible. So, in that case, you pretty much have to wait until you know you'll have time to play as thoroughly as you want. I've never tried, but I don't see why not. It deletes your save when you die (and I think it only allows one run to be saved at a time, so starting a new game will delete one that's already in progress...but, again, I've never tried this), not when you quit (unless you click "End Run"...). It's pretty much Deus Ex as a zany top-down brawler/shooter with Rogue-like elements. There are always multiple ways to accomplish any mission, though some are limited to characters with certain equipment or traits. There's a number of systems that interact with each other in interesting ways. There are a bunch of different characters you can play (several available from the start, plus probably two or three time as many that can be unlocked by achieving certain conditions -- many of which are quite easy to do accidentally), and many of them play very differently from one another. It will honestly be a long time before you'll be able to say you've seen and done everything in the game.
Now, I am admittedly a sucker for certain types of run-based games with procgen or randomization, so I don't fear permadeath or a certain amount of repetition. But even taking that into account, this is easily one of the best "roguelite" action games I've played. It's also one of the most customizable.

With regards to the disasters -- yes, the third and final level of each of the five main districts of the city (that is, the game world) has a random disaster drawn from a pool of something like 8-12 possible ones. Some of these change the game in interesting ways, some are just annoying (and which are which is probably going to differ from player to player). But the big thing about them is that they can all be turned off.
The full game has "mutators", which are like in-game mods that you can use to help tailor the game to your tastes (to an extent). There are a bunch unlocked from the start, and more have to be unlocked through meeting certain conditions. I know there's an individual mutator to disable each of the disasters individually; I think there might also be one to just turn them all off. (And, if you're masochistic, there's one that turns them on for every level!)
There's also a mutator that makes the game shorter (by removing some levels), and one to give the player some continues (like in old-school console/arcade games). There's even one or two "cheat" mutators.

Aside from the normal exhortation to "watch some videos", I'd strongly recommend you try the demo, if you haven't already. It's pretty old (I think it's basically a slightly-updated early access build), but those first 6 levels that you can play through in it should give you a good idea about what you'd be in for if you buy it.
And I'd say it is totally worth $10. But even if you're poor (or a skinflint), it's a steal at much under $7-8..assuming you like the demo. (I got it for about $5.50 in the winter sale, and regret it not at all, unlike some games I've gotten for the same or less in about the last year.)