Has anyone come into contact with owners that want you to Stay the night while they are home?

I have only once ever stayed in the house when the owner was home. That was only because they ended up getting back super late in the night, and I lived far away. Ask if they have a particular reason they'd like to do it that way. Maybe they want to make sure you're there before they leave (I've had this happen before). If this is the case, offer to meet them before for a meet and greet to build trust. This lets them see how trustworthy you are, so they might not need you there the whole night before they leave. If they aren't persuaded, it will give you the chance to assess if they feel safe, the home provides the security you would need, etc.

If you really don't want to be there overnight when they're home (I wouldn't), just let them know that you're happy to try to accommodate early departures, etc, but that for security reasons, you would not be comfortable making this arrangement with first time clients.

I've had an owner offer for me to stay the night when they weren't getting home until super late (almost the next morning), but even then I declined as I think that's awkward.

Personally, I would offer to come early in the morning, but you can do whatever feels comfortable to you. Don't be worried about denying requests that aren't a good fit, although rover typically sends us good people there are a few weirdos and weird situations out there that should be avoided.

Very odd. Stick with your gut.

Even if they are leaving early on Saturday, there's no need for you to stay overnight prior. Offer to come early. They can let the dog out briefly before they depart and then you can take care of its normal feeding and walks once you arrive. Do whatever feels right. Trust your own good judgment.

I wouldn't do that, seems strange