Am I allowed to sign up for all the credit cards just to get those cash rewards and then cancel right after?

Yes you can do that, but be sure to read all the fine print and t&cs before doing so, and do only 1-2 cards each time so it's more manageable and realistic to meet the spending criteria also.

You probably need to fulfil some conditions like some spending with the new card, or hold it for another 6 months or so before allowing to cancel.

But try not to do this with too many banks all the time, as you might be negatively affected in your credit score, or other matters within the banks.

There should be some requirements for you to fulfill before the rewards are granted. No free lunch.

If your application is approved, just keep the card till the bank charges you the annual fee. Don't need to cancel it prematurely. Banks like to serve customers who spend, not card hoggers.

Yes you can. But usually there's a some time before you can cancel, either 6 months or 1 year. To sign up again and be eligible for rewards again, there's a 1 year period usually. But some companies would put you under blacklist where they let you apply, but won't give you anything even if you meet any of the criterias.