These are all correct. We also use the verb 'to brake', a lot, as Jordan has used it. I would say 'I had to brake suddenly', or 'I braked slowly to avoid skidding', or "Brake!! Brake!!' (to a learner driver). 'Apply the brakes' (or its opposite, 'release the brake') is the sort of thing you read in an instruction manual. Pumping the brakes means putting your foot on and off the brakes quickly, to dry them or to avoid skidding.

There are many. In the UK we normally "apply the brakes". You can simply "use the brakes". For an emergency someone might shout "Hit the brakes!" In a race, the driver may repeatedly "pump the brakes".

You can also "apply the brake". "Step on the brake" is used, but I think it's mostly used when someone brakes quickly. "Press the brake" and "apply the brake" have less of an urgent feel. The driver is probably more calm.

Out of the options you've provided "step on the break" is the most common. However I think "press the break" is the most common of all.