re: the phone number
To be fair, I keep my out-of-state cell phone number because it's the one I've had for a very long time. I don't think the area code should be any factor. Also, if you're using Rover's provided numbers, mine doesn't match my cell phone or current state!
re: meeting a stranger
If you are not comfortable, I'd suggest having a family member and/or friend at your home with you at the time of the Meet & Greet. I try to always schedule mine so that my husband is home with me. Alternatively, you could meet somewhere near your home, perhaps the corner coffee shop, and then if you feel safe, have the dog owner follow you home.
re: Rover account
The fact that a person has contacted you on the site means they must have an account with Rover. Click on their name and make sure they have their dog(s) listed. I've had clients who are first-time Rover users, so they don't always have complete profiles. You could ask them to upload some pictures of themselves and their dog(s), if that helps you feel a bit better about them!
Good luck! I know working with strangers online can be pretty scary. Just remember to keep yourself safe first, even if it means a little bit of inconvenience for your clients.