What rights do I have during the purchase process? Can I ask for a copy of the tenants lease, tenants ability to pay rent? Do the existing terms of the lease to the renter transfer to the new owner? How are deposits previously paid handled?

I'm in the same situation - new to investing.  The Realtor stated the tenant has 60 days after purchase to vacate?  Paying way undervalued rent. 

@Jan Vandenberg During the 10 day inspection period (when utilizing the AAR contract) you can request any document that are pertinent to your purchase.  If you don't get the responses or information you expected, you can cancel the contract and receive a full refund on your deposit.

Yes you can pickup the lease if there is still one and have the seller transfer the deposits to you at closing.  More times than not, the tenants are on month to month leases which may or may not be in writing.  That's when its very important to do your due diligence and have the owner provide proof of rents and find out about any deposits outstanding.