Schultheis is a small neighborhood consisting of large homes on a quiet cul-de-sac off of S. Peninsular Dr. on the east side and the Halifax River to the west. It's comfortably close to the beach without the traffic of living next to it. Being in a resort area, there are a number of restaurants within easy driving distance for you to sample. There are dog-friendly parks and pet supply stores within driving distance for you and your pup to visit on a regular basis.
Being sandwiched between the Halifax River and the Atlantic Ocean, the climate here is spectacular. Start with winters in the 40s and summers in the 90s, add in the distinct lack of snow and approximately 230 days of sun, and you can't go wrong. The only thing you have to worry about is the 51 inches of annual rain, but you can always carry an umbrella!
Being on the beachside puts you in between the Atlantic Ocean and the Halifax River only four blocks from the beach. Unfortunately, there aren't any sidewalks for the two of you to go walking on, so be sure you are dressed to be noticed by motorists as you stroll, especially at night. Remember that this area is heavily frequented by tourists who come to enjoy the beach and the many shops along the A1A, so keep your eyes out for stray traffic.
The good news is that you can walk the four blocks to the beach and then go for nice long strolls together along the sand. Early mornings are an excellent time to do this as there are fewer people to deal with. Keep in mind that both the pavement and the sand can become hot enough to burn your pup's paws when the outside temperatures reach 70°F and higher.
There isn't much in the way of greenways here for the two of you to enjoy beyond your own yards. The greenways along the Halifax River are, for the most part, the backyards for the many houses built along the banks of the river. But if you don't mind taking your pooch for a car ride to the north, The Daytona Beach Lenox Playground has plenty of grass for the two of you to run around on, a walking trail that goes around the park, and a small dog park.
For the ultimate greenway to play in, you have your choice of Tomoka State Park, the Tiger Bay State Forest, the Relay Wildlife Management Area, the Bulow Creek State Park, and the Gamble Rogers State Park. All of these parks have plenty of long trails for the two of you go hiking along, seeking out the many forms of wildlife who call these parks home.