If you mean a fire just burning in the level, no. You'll need to jump over it.
1) The game only saves when you're in your waiting room. That's the only time you can quit the game safely. if you quit out while you're in the tower you'll lose your character. The game is designed this way intentionally. Elevators just act as a means to get you back to the waiting room more or less. The layout of the tower will change everyday however all the areas that run central on the map (press down on the d-pad to view the map) and fall along the bold purple line are static and any area on that line has an elevator. So your short term goal when pushing through the tower should be to find the next available floor that has access to the central elevator. This will allow you to return to the waiting room and effectively act as a shortcut to get back to that floor. Also be aware that anytime you leave an area (use an escalator or elevator) and then go back to it, enemies will respawn but materials will not. Bosses and materials will only repopulate when you go back to the waiting room.
2) Use R3 to lock on. Press up on the d-pad to switch between targets
3) Yes, you can use decals from the mushroom lady to boost your stamina and/or stamina recovery and as you get higher grade fighters, their attribute caps (HP, strength, stamina, dexterity, etc) rise as well.
4) Believe it or not, the game is not very intrusive with it's F2P stuff. The only things you'd really spend money on is for death metals which let you revive instantly when you die. But there are ways to get death metals in the game plus you want to be smart about how/when you use them (using them on bosses should be the first priority) The game is called Let it Die for a reason. The other thing is using your death metals to expand the storage slots in your chest in the waiting room. those slots fill up quickly as you gather materials and eventually you'll want to expand you're storage space. So basically, the F2P stuff is more about convenience than necessity. You can also buy the express pass which gives you nice benefits like extra death bag slots, free elevator usage, free daily decals and a few other things. Again, it's more of convenience rather than necessity.