I basically avoid having to deal with them electronically, but I still collect and dispose of them offline.

I think the first logical upgrade is the lighting and creating electronic and trackable access.  

Be prepared for them to only have electronic payment too.

The biggest thing for STR is having reliable cleaning crew, everything else is electronic and can be done remote.

Would going through one of those electronic notary platforms allow me the ability to say I do business in X state?

In some states the laws on eviction can be effected when you allow tenants to make deposits, since you can not prevent them from making a partial payment.I use IPN which is part of quickbooks for electronic/online rent payments.I will also highly recommend that you do move in walk thru inspections with your tenants, this will give you an opportunity to answer questions and discuss the difference between damage and wear and tear with your new tenants.

They can be sent back and forth via fax or electronically for the most part. 

I mailed my payment (for March) in a week ago and they processed the check electronically yesterday.

It was mostly (stolen) power tools and electronics.

Recently I set up all of my tenants on the quickbooks IPN payment system which allows them to pay rent electronically for a fee of $0.50 per payment.