You build your profile (lots of personal details, address history, ID details blah blah) and then you submit the 'training request' online, obviously for initial most likely first. Fingerprinting, as long as you keep your profile/name/doesnt get lost in transit before digitised, is 'good for life' for this TSA thing.
Speed of processing depends on where you do it. YOu can do it in few places outside USA, but that's at premium cost. If done at local sheriff's/police&court facility, it's normally within day or two, as long as you got your online side of TSA done. It is bit of question of 'piece of string' length to give specifics.
Regarding starting flying before the permission, well, I'm sure not all adhered to it. OTOH, 'demo'/trial lesson x10 is perfectly legal, as it's not training course per se. You can also hit the self-study, get over jetlag, do bit of socialising etc. If you're on whole zero to hero course, day here or there won't matter much.
Stop worrying about 'week and not in hand yet' I-20. You are aware admin can take day or two, then snail mail from USA to wherever you are. What you can do, is register on TSA website and have all your details there ready, so you have fingeprints done and request filed at earliest opportunity.