Allow the tenants to take possession but defer payment until June. 2.

A lease gives the tenant possession of the property for the term so long as they abide by it's terms, if they tell a good enough story and they have a right of possession the judge is way more likely to give them extra time than if they are hold-overs (They still have to pay though).

Once big issue is when they move out, they claim a deposit was given but you never saw that or got that $$ from the previous owner, so now you are stuck etcGet a copy of the move in/move out from the original owner.....and then do a walk through when you take possession to document it now.

They were moving to Europe and selling most of their possessions.

If it was a screaming deal, I would find a carrier that would cover me in the short term, take possession, then immediately terminate the tenant's lease. 

The unfortunate thing about this situation is I took over possession on the 10th.

He also informed me that the tenant involved had been arrested for drug possession and is now jobless.

The landlord then documents their finding of the abandoned apartment, including evidence of the tenant's abandonment, such as no rent for the last month and no possessions left behind.

Despite the lease, does she at some point get tenants' rights (I'm thinking something akin to "adverse possession.") or do I have to option to consider her a trespasser if I choose?

Explore smaller multi-family units for robust cash flow or venture into long-term rentals and flipping if you possess construction expertise, though the latter involves higher risk and effort.