Yes, day trading is allowed on both the long and short side. They will even let you short the financial stocks again.
Not sure the minimum. I think you can open an account for $500 or less. But if you want to short stocks, you will need a margin (credit) account. This can be somewhat of a pain to get. Probably easier to get with a large initial deposit.
The other way to bet against stocks is to buy PUT options. Costs a lot less, but is more risky and volatile.