Hi Summer - here's my experience as well. I mostly rent for 3-6 months at a time to families relocating from out of state, so some of my answers may not be as applicable to travel nurses.
Tenant screening: I do a background/credit check, but I don't ask for references. Since my tenants are usually homeowners it doesn't make sense for me, but even for the couple travel nurses I've had, the background/credit check was sufficient and seeing how they communicate with me prior to booking. Some MTR hosts will also skip a background check for travel nurses in lieu of receiving their employment contract. Travel nurses go through extensive background checks to be employed, so having a copy of their contract is sufficient for some. Make sure you feel comfortable, but keep in mind if you ask for too much (in comparison to other MTR hosts), you might lose out on guests.
Income verification: I don't ask for it. I've found that by requiring a deposit, not being a low priced rental, getting back a good credit report, and the fact that they're relocating to buy a house it's not something I need to verify. For travel nurses, you can ask for their contract, but even then, I've told my travel nurse guests they can black out their pay. You can call the contact on their contract to verify employment. Usually, a red flag for me is if anyone asks for a discount.
Leases: Each tenant should complete their own background check, application, and lease. I don't charge an application fee, but they pay for the background check. Since MTR stays are a month plus, guests will have tenant rights and need a normal rental lease. I got mine off Law Depot. I wouldn't worry about the length of it.
Pets: These aren't short-term guests, so for me, they're allowed to have their pets as they would in their home. They can be left alone, or they need to set up boarding. I wouldn't want my guests to leave their dog outside all day. I confirm prior to booking their pet is house trained and is a minimal barker. Usually if they're upfront about their pets they're fine. ESAs aren't a red flag for me, esp if you allow pets.
Rent: If they move in in the middle of the month, I pro-rate it and it's due prior to them moving in and me giving them a door code. Beyond that rent is due on the 1st.
Trash: Yes, include trash expectations in your lease.
Hope this helps!