Forced to pay 1 full month rent for August, but can't move in until 8/22. Is this right and Is there something we can do about it?
Not if that's what is required,and you agree to it, it's a security deposit.
Technically then your rent will be due on the 22nd of the month. They can't make you pay for a whole month if you move in that late. See if you can reason this out with the landlord. In any case save ALL your rent receipts.
You cannot be forced to pay for rent tin a house you don't live in. If you are you pay your rent on 22nd, that means you will pay again on the 22nd of the other month.