This Holding Deposit will be applied to the non-refundable and security deposit when Applicant takes possession of the above-described unit.

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?

You are about to turn over possession of your property.

Also, many states are in fact legalizing weed for personal or medical reasons, but there's still limits on the amount you can possess at any given time..

You can also do a pre-walk through to gauge condition.If you don't have time to fix something, either fix it after the new tenant moves in or document condition for when the new tenant moves out.If you approve the application, just make it clear to her that her deposit will be returned based on the condition of the property and is subject to the new tenant taking possession as agreed.

Your goal at this juncture is to reclaim possession of your property with the least amount of damage. 4.  

And they are looking at taking possession in about a month.

Why wouldn't the owner just push back the close date or let you know they would be there when you took possession and then move out in a month? 

You should complete the eviction process to be entirely sure you regain possession legally.