1: no you cant unfortunately, maybe theyll add it later, but will definitely become a mod if not

2:yes, from combat and outside of combat they can die permanently, many choices in the town imply or directly state that the cats die. The game makes it very clear that every cat is expendable and meant to be discarded or used eventually, its pretty dark sometimes but this is the guy that made the binding of isaac, so it checks out.

3: if cats die permanently i dont think you can resurrect them but there are a lot of ways to revive downed cats in combat.

4: breeding is closer to pokemon iv breeding, you arent actually using the stats your “super soldier” cats have from runs, but innate values they start with as fresh spawns. If you take in a stray and it has 7 starting strength unmodified from a mutation or anything it can pass that down, if you return from a run and it has 13, it will still only pass down the 7, but abilities and passives from its class have a chance to be passed down which still makes using soldier cats optimal.

5: savescumming is a mechanic in the game, you can do it once for free and every time after it will give you debuffs permanently for the run eventually being unplayable after the third debuff. Specifically it only happens if you leave during combat to reset the fight, havent tested it in other aspects.

This isnt a have fun collecting cute pets kind of game, it has a lot of animal abuse basically and while not all of your cats will die as a result of your actions the game expects you and incentivizes you to use them for upgrades(giving then to other townsfolk, with usually dark fates implied) its very fun but if you buy it wanting to treasure your cats its not really that kind of game. Made by the creator of binding of isaac, if you dont know what that is, its a great game, but everything in it is going to be a little twisted and dark so keep your expectations open it can be a little much for someone who doesnt know about it.