Is it legal for Apartment Managers to demand payment for a 60-day notice while not allowing us to live there?

Your cousin owes this, not you. Your mother can't just move into property like that, she has no legal right to be there. The rent is paid by the cousin, but that does not give your mother any rights at all.

Well your cousin is the one on the lease not you or your mother. That is the problem, so yes in Virginia you do have the right to bar guests from the property. So really it is not our apartment lease it is your cousin's apartment lease and you are only guests. You should have obtained permission before just moving into the apartment. About the only thing you can start doing is start packing or make them evict you, which would not be hard since you are not legal tenants.

EDIT: After re-reading your question, I'd like to ammend my answer.

You said you did not sign the lease, only your cousin did? You have not signed any contract with them? You will not owe any fees to them over and above the last month's rent. Your cousin should not owe any rent or any "60-Days Fees", as rent was paid through the end of the lease (unless te contract/lease he signed states if he sublets, he is charged a fee). Your cousin will be liable for any damages to the apartment over and above normal wear and tear, so make sure you clean it real good and make any repairs needed, as well as clean the walls with soap and water (or wall-cleaner) and steam-clean the carpets. Do all this as close to move out as possible, preferably after it's emptied. Keep receipts for all cleaning products and/or services! Give copies of receipts to your cousin as soon as possible. Hopefully, he will be able to get his deposit back.

Good luck!

If they demand you leave then leave and notify your cousin the reason etc. They will charge him for breaking the lease unless you can get them to sign something to let you out of it.

Explain the full reason you had to move such as they said you made too much noise etc etc

i think of all the different solutions are actually not understanding you question. in case you have a hire you're actually not required to supply observe in case you should pass on the final day of your hire. You did not even ought to supply him the 30 day observe which you probably did. That replaced into superb of you yet not required. And no you do not owe the months hire after your hire is up.

You cannot pay rent for a place which does not allow you stay there. Contact your local authorities. This is illegal.

You or your your cousin pays. The landlord is entitled to that money

Yes. You owe the money. You already declared when you were leaving. So that's on you.

illegal, abosulutely. you need to contace your local authorities.