When I updated to WoW 5.4.3, all my battle pet addons broke.
I am using the latest PetBattle Teams, PetCaught, PetTracker, Pokemon Trainer.
What happens if I have any of these loaded is:
(1) My pets, both battle pet and combat pet, (I am a hunter) do not follow me around: They are in-game, but just standing in the spot where they were when the game launched.
(2) My 'logout' and 'exit' system menu buttons do nothing. I have to manually kill the game from the Windows desktop to remove it.
(3) If I try to dismiss my combat pet, I get some animation, but no progress bar and nothing happens.
If I dont load any battle pet addons, the game plays normally (and all my other non battle pet addons work). If I load any single battle pet addon, or combination thereof, I get the above happening.
What you describe looks like severe network lag, but I am not aware of any addons that caused such lag. Maybe Blizzard introduced a bug with latest patch that is triggered somehow by pet addons. Anyway, as Elkano suggests, try to see if any addon is causing lua errors.
PetTracker causes my game to lag really bad, when I turn it off the game runs perfectly. I noticed that it is constantly climbing in the amount of data it uses, which is why I started with it when trying to see what addon was causing my problem. Also, PetTeams is not working at all. No option to even make my teams up or use my pervious set ups.