Quick rise is something referred to specifically as the straight stand up, or pushing down. Back roll/rise is when you push back.
The timing window is relatively loose, just push the button whenever you hit the ground or slightly thereafter. It can get tricky with some throws that have a different timing to them, but you can just mash if you really want to get the quick rise.
Quick rising is something you can almost always do in SFV. There is a halfway between a hard knockdown from most games and what there is here, which is following throws and command throws you can only do a quick rise, but not a back roll.
A full hard knockdown does exist in a few situations, most notably after a crush counter sweep. Some very specific moves only offer the same singular option of auto standing like M. Bison's EX Headstomp or his EX Scissor kick.