L.K.
If you are more uptight about hair, I'm not sure a golden is the right dog. They shed, A LOT! Our cat used to shed a ton too.
While you ask great questions and you are asking for help, do you realize the amount of work it is going to take to train a new puppy? With 2 little kids? And sorry, even the best trained puppies will have accidents, may chew on things even with toys and bones available. Are you going to have the cat declawed? Some cats will still scratch no matter what.
Sometimes they will bond, sometimes they wont. We've had both situations.
And just like kids there is no guarantee to health. Our 5 year old golden/lab mix got sick last fall. A bizarre and very rare issue with his immune system. Because he was young and had seemingly been the picture of health, of course we did many things to bring him back to normal. Which cost us $2500+
Big dogs eat a lot. I have 2 big dogs and we go through a 35 pound of dog food about every 3 weeks. Because I am as picky about my dog food as I am my people food, it isn't cheap.
Personally, I would go to a shelter. Or go to a rescue group. Many times they will have full blooded breeds. And even if you get a dog that is a few months to a year or two old, they may already be trained. I swear they realized they are getting a second chance at life and don't want to mess it up! My last 5 dogs have all been golden rescues. Our cat too was a rescue.
Good Luck