For those who love the city but do not necessarily love being a part of the noise, North Queen Anne provides the perfect compromise. Part of the larger neighborhood of Queen Anne, the northside of this hill is quieter. You still have easy access to the main part of downtown, but you also have access to Fremont and West Woodland over the Aurora or Ballard Bridge. This greatly increases your access to amenities for both you and your pup.
While there are no pet stores in North Queen Anne, there are a number of places to go for grooming, food, and medical care around the greater Queen Anne Area. There are not a lot of businesses in this neighborhood and the few that there are exist along the canal and around Seattle Pacific University.
Weather in Seattle is generally amenable to walking year round with short breaks to avoid the Pacific storms that come through.
North Queen Anne is the north face of Queen Anne Hill which is the tallest hill in Seattle. There is no part of this neighborhood that does not have a rise in elevation. Many of the roads have very challenging hills that may be daunting for the new dog walker to tackle. If you plan to walk in this neighborhood, it is advisable to drive through it first, marking out a route that feels manageable. There are a few that are not as difficult as others, so you may be able to fashion a route that works for you.
Your other option is to head down to the Fremont Cut. There is a trail that runs along the canal that will be much less overwhelming for you to navigate. The entire neighborhood, luckily, has sidewalks, but the residential areas do not have lights or other traffic control because they are quiet enough to not need it.
While North Queen Anne does not have a lot of greenway spaces itself, there are some good options available for those who want to get off the concrete while walking their dogs. David Rogers Park is a popular place for residents. This park is not quite as steep as the rest of the neighborhood, so it makes a good place to go for pups who are not up to the hills of Queen Anne just yet. Be aware that, even here, there are hills, so if you want flat terrain, there are better options.
The South Ship Canal Trail is a good option if you are just starting out. Along the canal, the terrain is flat, but there is still plenty of greenery to give your pup a nice place to explore. There are places to park, so you can drive down the rise and park near the canal to take your walk.