Personally I wouldn't do it. I'm seeing red flags. But maybe they're just clueless to how the system works and need you, the professional, to walk them through the steps. When we do meet and greets, I have the potential clients come by our house, but I meet them out at the street. Then we take the dogs for a short walk to get acquainted, and if the dogs are getting along well, there's a field at the end of the block where we can give them a little more freedom to play. THEN, I will invite the potential clients to come inside and see where their dog would be staying.
If I were in your position, I would request to meet somewhere safe and public first, like a park or coffee shop so that you can chat with the clients and (if possible) meet the animals. After getting to know them a bit, I'd be more comfortable visiting at their house, but only with a clear time limit. Still, make sure someone knows where you went and who you are with, and maybe have a friend drive you to the meet and greet and wait in the car.
I'm not sure from your question, but it sounds like they're wanting you to stay there while they are home, and quite possibly without compensation? That's not something I would go for. You're a professional and your time is valuable. I'd encourage them to ask you any and all questions they'd like about the type of care you provide during the confines of the meet and greet, and to message you if they have any further concerns, but staying overnight is a service you charge for. And of course, at the first sign of anything creepy, excuse yourself and report the encounter to Rover.
No, absolutely not! If it was nice old lady that was just worried about her pets, you could consider it. However, you would have to be paid and covered by Rover Insurance. It is a very odd request. If there is anything that makes you uncomfortable don't do it.
If it were me, I wouldn't even book with them. This sounds sketchy. *Shudder*