That is NOT a household employee. A household employee is like a nanny or persons that provides services in yoir home.
No. A dog walker would not be considered a household employee. Per the IRS, a household employee is if you hired someone to do household work and you could control what work he or she did and how he or she did it, you had a household employee. In addition, household work is work done in or around your home. Some examples of workers who do household work are: Babysitters, Cooks, Maids, Butlers, Drivers, Nannies, Caretakers, Health aides, Private nurses, Cleaning people, Housekeepers, Yard workers, and etc.