As the wife of a SWO, there are times when it sucks big time and times when it is so incredibly awesome words cannot describe it. For the most part your family will be able to move with you to various duty stations. I don't know what the SWO pipeline is like right now with regards to BDOC, SWOS, etc etc etc and so I won't comment on those. However, in and amongst those initial SWO courses you will be assigned to a ship for your DivO tours. Unless you get assigned a ship OCONUS and your wife/kids have a medical issue that would prevent them from being able to move with you (see references to EFMP, overseas screenings, and unaccompanied tours), they will travel to your duty station with you. That means that when your ship is in its homeport, you will be going home each night* (except duty nights and nights where you would get more sleep just staying on the ship e.g., INSURV prep) to her/them. When your ship is deployed, unless she/they manage to fly out to meet the ship in a port (it can be done but is a bit of a logistical nightmare. I did it once and that was enough, lol.) You won't see her/them until the ship returns home (see above "incredibly awesome words cannot describe it") While you are gone, those at home go on with their regularly scheduled lives: jobs, school, volunteering, housework, etc etc etc. Depending on the person and the duty station, sometimes it can be a bad experience. All wives have their own 'sea stories' of a duty station or deployment where life was pure and utter hell while the ship was gone. It isn't always a bad experience and quite often deployments can actually be a good thing for spouses! I know that I made several friends from the FRG/wardroom that I might not have met had the ship not deployed and I got active in those groups. Friends definitely make the experiences better!
Speaking of friends, it is a VERY small Navy. After a couple of duty stations you start to run into people again and again. For example, TrainO and family from USS Secondship of PenguinGuy lived 3 houses down from us when we moved to Monterey. We will all overlap in Newport for a month or so before they move to Norfolk and then we meet them there again several months later. SuppO from PenguinGuy's USS Firstship ran into him at NPS a couple of months ago. Don't even get me started on how all of my closest Bunco playing friends from San Diego came through Monterey and inevitably we will meet up with them down the road.
As a spouse it is important to be flexible and openminded. Oftentimes you hear horror stories of a duty station, but it is very much what you make of it. Your wife will be fine, there is a LOT of support out there. Heck, if you get selected for SWO let me know and I can add her to the SWO Spouse FB group. I am also available for particular questions or if your wife wants to talk herself.
As for your dog? Don't have one but aside from logistics of PCSing (i.e, breed restrictions, overseas quarantines, finding pet-friendly rentals, expensive airfare overseas, etc etc etc) there is really no reason why your pup can't keep going where the family goes.