When doing dog boarding, is there an option to start a walk like there is in doggy day care or track pee / poop / etc?

No, that function is not applicable to boarding or house sitting.

I am so thankful you asked this question because I could not find a Rover card option at all. I am used to doing drop in services for clients. I was very disappointed that I could not complete a Rover card for overnight boarding for a new client.