When dog owners are searching, does my profile rise to the top of their search based on whether I have taken care of that breed?

Just a luck of the draw

No, Rover's search function doesn't allow them to search by breed, only by size of dog. It could be coincidental that all the small dog owners happen to have Yorkies.

It doesn't search by breed but yes, what you sit for in the last the algorithm will send more of the similar kind to you. For example if you watch puppies: you'll get sent more requests for puppies because "you're good at it" etc.