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.

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

Not for their personal possessions but for the damage they caused.

It also makes it one step easier to evict them (if that becomes an issue) than with a standard lease agreement.Regarding your tenants, just have them sign your new rental agreement the moment you take possession. 

If the tenant has lived in more than two places within a two year period, it may indicate instability so I would require a previous landlord reference and make sure he actually owned the property.Whether or not the tenant's current or former landlords release information, we landlords are responsible for who we allow possession to our properties.Hope this helps!