While I too am leery of last minute requests, I know there are certain situations where pet owners may be scrambling to find care last minute. For many people, plans fall through and they find themselves having to use Rover with rather short-term notice. Because I have my own dogs, I do not accept boarding requests without a M&G.
For last minute boarding requests, I ask the pet owner to meet me at the dog park or the green space of my apartment and let them know that I will gauge compatibility with one of my dogs present. I let them know that if it's a good fit, we can confirm the booking on the spot. (*Note: I do not take the dog from the owner until I see that they have paid through Rover). If they are not willing to do this, then I give them the number for Rover Match, so that Rover can help connect them with a sitter that is available to take last minute requests.
For last minute walks, house-sitting, or drop-ins, I am flexible with last-minute requests. I'm happy to meet the owner before they leave to pick-up a key or have one left for me under the mat. I do let them know, though, that if their dog doesn't allow me to enter (aggressive/territorial behavior), that they will not be refunded for the attempted visits.
In either circumstance, I ask the owner to take 5-10 minutes to update the pet's profile on Rover so that I can easily access any important information on routine, medications, food, etc. This is something that I also mention at meet and greets for other clients as well.
In any case, always be cautious when gauging new clients. Even if it isn't a spam request, it is the sitter that is assuming a lot of responsibility (and liability). It sounds like you are doing this already, so you're off to a great start on Rover. I think that other sitters would agree: if it feels "off", it probably is. The longer you are on Rover, the easier that you will be able to discern whether a last minute request is truly an emergent situation, a scam, or just the red flag of an irresponsible pet owner.
always go with your gut feeling. The first time I had a request it felt iffy and i didn't do it.