Step 1: 8 moves (6)
I've found positions where each corner requires 8 moves, but chances to get one after random scrambling are purely theoretical. The chance that all corners require at least 7 moves is very low as well, less than 1/10000. So in practice, you can trust that 6 moves is enough.
Step 2: 9 moves (7)
It's possible that all 3 options require 9 moves, the probability is 1/21946982400. And 8 moves is rare, too, I believe it's less than 1/100000 (but I haven't checked yet). So again, in reality, you won't need more than 7 moves.
Step 3: 8 moves
Color neutrality doesn't matter here, worst case is 8 moves. With practise, though, you'll be able to look ahead during the pevious step and you often get a cancellation.
Step 4a: 10 moves (8)
Probability for 10 moves is 1/146966400 and 9 is rare, too.
Step 4b: 9 moves
Color neutrality doesn't help here, either; worst case is 9 moves.
LL
Move count is not the most important thing here in speedcubing, and there are many different ways to solve LL. But using optimal ZBLL, the worst case is 16.