Hi Simone,
That's great that your current resident is willing to share the bathroom...sounds like you have a good one! I'll address your questions from my perspective in order:
First - You sign a lease agreement in addition to the airbnb booking. I do this with all residents who book beyond a 28 day rental through the platform and have it mentioned in my listing and rules before booking.
Second - As for marketing, if you're using airbnb, that's your marketing. I personally also post my MTRs on Zillow and specify the length of time available and that it's furnished. You could also share it with local corporate recruiters, hospitals, military housing sites, etc. For screening, I use either Zillow's application process or if not coming through Zillow, apartments.com to run the credit and background checks. This is a free service for you as the landlord. Separately, I use apartments.com to collect rent, security deposits, bills, etc - also free for you as the landlord and the tenant.
Third - to write off items for taxes, I'd check with your CPA but in general, it would be anything you spend money on with that property, including mileage going there.
Best of luck!
Lindsey