Useful on any terrain where traction may be a concern, cleats are shoes with spikes or hooks attached to the soles. Cleats reduce the penalty for walking over slick surfaces by 50%; for example, walking across ice normally costs 2 squares for every square of movement, but with cleats it costs only 1.5 squares for every square. Cleats cause damage to any type of finished flooring.
If I understand this correctly Cleats movement seems to work like when you wear heavier armor
1) Can you 5-foot step?
2) If you move 1 squares how much movement have you expended?
3) If you move 2 squares how much movement have you expended?
5) If you move diagonally 1 square how much movement have you expended?
6) If you have a 30-foot move speed can you move 20-feet (4 squares)?
7) If you have a 20-foot move speed can you move 15-feet (3 squares)?
8) If you have a 15-foot move speed can you move 10-feet (2 squares)?
9) How does this work against spells like Grease?
1) The cleats don't change difficult terrain (due to slipperyness) into NOT difficult terrain (they only partially mitigate the penalties), so you still can't five foot step. Cleats aren't described as having any effect other than damage to flooring when NOT on a slippery surface, so they have no effect on five foot stepping when in normal terrain.
2) 7.5 feet.
3) 15 feet.
4) There is no four.
5) 11.25 feet. However, due to the way Pathfinder treats the first such move as 5 feet and the second as 10, it would be 7.5 and 15
6) Yes.
7) No. After moving 2 squares, you've expended 15 feet of your 20 foot move speed. This leaves you only 5 foot remaining, so you can't pay the 7.5 feet you need to enter the next square.
8) Yes.
9) Grease calls out no exception for cleats, so that would be GM fiat. Now, if the grease was not cast on an area of floor but on the cleats themselves, we're outside of RAW.