On average, the weather in this area is never too extreme and remains in the mid-forties to the high-sixties range. Several bars and restaurants offer dog access, just be sure to ask. The sidewalks are wide, so crowding shouldn't be an issue. However, there are multiple-block stretches without grass or medians, so pack clean-up bags and plan ahead to hit up a greenway at some point during your rounds.
Aside from the hills, the terrain is what'd you'd expect in a downtown environment in that it's mostly urban with little differentiation. However, this area does offer up the option of a bayside stroll or a visit to the park. Along the sidewalks, there are trees sprinkled in, so don't expect a walk devoid of all things natural.
If you're on the south edge of FiDi along Market St., your closest spots are the Yerba Buena Gardens, heading away from the Bay or Sue Bierman Park and right by it. Make sure not to take doggos into the actual gardens though as this is not allowed and in other areas, they should be securely on-leash. Sue Bierman is a 5.3-acre park outside of the high rises, offering open sky and ocean proximity.
Lastly, Sydney G. Walton Square is a cool area worth seeing. Surrounded by awesome architecture and sculptures, this little plot is near the top right corner of the district. Unfortunately, there are no leash-free spaces in the area.