Used to. Not anymore.

not sure how big is your backpack..
don't go those dodgy type of bathhouse. overall bathhouse in korea is quite safe. don't say broke the locker la, hardly got people steal your phone even you charge your phone at another corner. of cause i can't guarantee 100% no pickpocket however the percentage is quite low.. 

I think should be just enough to squeeze into the locker. Honestly if cannot, just put it next to you where you sleep. I've seen people putting their huge-ass luggages next to them while they sleep too.

As long as you don't go to those jjimjilbang frequented by the local gangsters, you should be very well taken care of in terms of safety. Not that the "dodgy" looking ones are not safe, rather you just may not be allowed to enter in the first place.

don't compare it to m'sia lolz. too way behind already. koreans can put their bag in the restaurant or cafe and walk away to order etc. they can leave their phone to plug in to charge on the table (mix with other people phone) and the distance is far from their place. not to worry that people will steal it and etc    in m'sia if you do that, you can say   

you can ask the person there to keep your backpack if the locker is too small.