So if a centre piece popped out and I finished the solve and stopped the timer with the piece still popped, it would be a DNF?
That's not true!
If ONE piece with one sticker on it pops out, no DNF.
If more pieces with one sticker pops out, DNF
If a piece has no stickers pops out, no DNF regardless of amount.
EDIT: From WCA
5b5) If at the end of the attempt some parts of the puzzle are physically detached or not fully placed, the following regulations apply:
5b5a) If one or more parts without coloured faces is affected, the puzzle is considered solved.
5b5b) If one part with one coloured face is affected, the puzzle is considered solved.
5b5c) If more than one part with one coloured face is affected, the puzzle is considered unsolved (DNF).
5b5d) If one or more parts with more than one coloured face is affected, the puzzle is considered unsolved (DNF).
5b5e) Regulations 5b5c and 5b5d supersede 5b5a and 5b5b.
So if one center cap, or one 4x4 center piece pops out, and you stop the timer without fixing, no DNF or penalties.
If a corner pops out (how did you manage that?!) and you stop the timer without fixing, DNF.