Make sure you got her deposit transferred to you and get the prorated rent. 

I make my deposits just under the rent (helps with avoiding confusing the deposit with the last months rent) and if they want me to hold it, they pay that full amount.I really recommend you spend some time reading in the Rental Property and Landlording issues forum.

Changes of late fees, grace period, due dates, rental amounts, deposits and other material changes can only be made once the original term ends or is lawfully terminated.

Think of all the things that could go wrong and see if the lease addresses them: unauthorized pets or tenants, early termination, security deposit, lease violations, late rent, eviction, lawn maintenance, parking, etc.5.

If there is a judgement amount awarded to landlord in the judgement, wondering if landlord can deduct the judgement amount from Tenant's security deposit.

Think of all the things that could go wrong and see if the lease addresses them: unauthorized pets or tenants, early termination, security deposit, lease violations, late rent, eviction, lawn maintenance, parking, etc.5.

Whether it's a pet deposit, increase in rent amount etc.Thoughts?

Because we didn't verify rent deposit at acquisition, we had to evict almost everyone after purchase.

(Anyone can make a deposit and she could write checks or use her dads credit card to anyone not taking payment in person.)

Yes the money placed in there is called the Earnest Money Deposit.