My advice would be to start with Spanish and then if you see you'll be staying in Barcelona for a while, you can learn Catalan as well so as to integrate with the locals more. Everyone in Barcelona speaks Spanish, so it is absolutely enough to be able to communicate. True, some official things may only be written in Catalan, but it has never been an issue for me (I'm a foreigner who speaks only Spanish, although by now I understand most written Catalan as well).
I advise this because you will most likely travel around Spain and Catalan will be of no use then. Learning Catalan has its nice added benefits, such as being able to get closer to the true locals, who do prefer to speak in their language (altough it's not really true what some will tell you, that Catalans won't speak to you in Spanish, it has never happened to me in the 7+ years that I've lived here). But if you're starting from zero, I'd say Spanish is definitely more useful.
As for the dogs, I can't really help much, as I've never had one. I'd say Barcelona is definitely a dog-friendly city, as I see a lot of dog parks, there's the beach, the mountain, etc. Finding a flat might be tricky, cause some landlords don't allow any pets, but again, I'm not sure how extended this is, cause I've never had a dog myself.
Hope this helps!
Good luck with the move, hope you enjoy Barcelona, it's a wonderful city in many ways!