After a stay, do you ask for reviews?

Yup. I request it. I simply say something like, "thanks so much for letting me sit for you. Please complete the review Rover sends. The feedback is helpful".
:)

I have started asking...which is a bit uncomfortable, but folks seem willing to write one after the request.

Absolutely. When I was first starting out I was a bit more on the ball about this. Now with nearly 100 reviews, I don't follow up as much, especially with repeat clients (more reviews don't necessarily help your search rank, but they work as nice references for owners browsing your profile). A nice, polite message always worked - "Thanks again for choosing me to host Fido! If you don't mind, please take a moment to leave a review about your experience."

When clients pick up their dogs, if they are new clients I tell them that Rover will be emailing them a request in a few days to review their dog's stay. I tell them it would help grow my business and I would appreciate it if they would write a review for me. If I don't see one in about a week I send a message like Sarah's.

For returning clients, I don't worry so much unless they didn't leave a review the first time. In that case, I remind them of writing a review.

Yup when I was just starting out I would text them a few days after the stay and say something like 'I hope Sparky is doing well! I would greatly appreciate it if you could review the stay; Rover will send you an email asking you to rate the stay. Thanks so much and have a great day!'

Just be as polite as possible and I always wait a few days after because Rover will send them a few reminders; the day after, a few days after and I believe a week after the stay so I always give them a chance to find the email on their own first. And obviously don't request a review if it didn't go well /if you told them you van't take the dog again