So long as you make sure that you acclimate your conure to a variety of situations, like you would a dog, I see no problems taking your bird with you wherever you travel!
As for pets getting along... no one can tell you. Some people have pets that completely ignore each other (such as my dog who'll ignore the birds, for the most part, and at best, they get sniffed - whereas the cats enjoy watching the birds, particularly the smaller species cause they are more flighty). With that in mind, someone lost their eclectus(es) to their schnauzers(?), who for years, had ignored the birds without issues... and one day, that changed! They can and do live in harmony without issues... but that's not to say it'll be the same for you. Then again, you never know if one day instincts will kick in. Having a well trained dog will certainly help.
As for suns vs green cheeks... well I don't own either, however conures in general tend to be nippy, especially the smaller guys. Sun conures also tend to be much louder than green cheeks, but that's not to say that green cheeks, in their own right, can't be loud, either!
As for cages, I would not recommend one smaller than 2' square, unless it's a travel cage/sleep cage/time-out cage. There's Flight Cages which I'd recommend, and considering their size, they are pretty cheap! (even with rising prices).
13221 HQ Flight Cage 32x21 - Presented by My Bird Cage
13221 Flight Cage
Centurion Aviaries & Cage Co. & Century Mutations (not as cheap, but they have the flight cage with seed guards, if you'd prefer that! and thus has it's perks over the others)
Otherwise, if you don't go for one of those cages, I'd recommend a cage that is 32" wide by 23" deep...
When choosing cages, be aware of fancy designs - they could pose a health hazard to birds... such as scrollwork, and anything that can create a "V" (particularly with legs, bars that make a half circle, etc)
Be sure to read up on a proper diet!!!! (which is, in itself, a controversial subject)
Oh, and btw, it's species, not breeds! So to speak, there are no breeds within parrots!