New landlord- Several ?'s on inherited tenants. Closing Friday!

[Brad E.]

Hi folks - I am closing on my first rental property (duplex) this Friday and have some questions about inherited tenants. 

1.  If I close on 9/1, which is also the day rent is due, how do I get the rent payment?  I assume the tenant would not be aware of the closing date and would submit payment to the seller as the normally do this time of month.

2.  Unit A's lease is up on 8/30.  How do I know for sure he intending to be out by 9/1 as the agents are telling me he is?  

3. Unit B has a lease extending through July 2018.  I want to have that inherited tenant sign a new lease with the same terms, but on my paper.  How do I go about doing this?  Can I force it?

3a. The current lease states that rent is due on the first of the month. This is in conflict with a section titled 'additional provisions' under which it states the tenant can break payments in two payments due on the 1st and 15th.  I do not want to allow this.  Can I remove this stipulation on the new lease I am wanting to have them sign?

3b.  I want to add additional fees, specifically for parking violations, so that I can use that as motivation should the need arise.  Can I add fees like this to the lease?

4.  I want to require renter's insurance.  Can I force the inherited tenants to purchase renter's insurance?

Thanks in advance for any wisdom you can share.