Hi,
We own fire arms as well, but they are put away safely from our ds. We are in the process of getting a gun safe as well for our handgun in the bedroom (for protection, just in case the bad guy makes it past the alarm system and the Doberman Pinscher). My sw wasn't too concerned about the guns.
The back ground check is mandatory and requires finger printing of all adults in the house hold. It takes about 4-6 weeks to get the results.
I wouldn't worry much about the animals. We have three cats (2 are strictly indoors) and a Doberman Pinscher, which as everyone knows, are killer dogs and eat small children for dinner (being sarcastic here). Shadow, our dog, is a big goof and while the sw was freaked out about his sheer size, she realized that he was a big marshmallow. The bm and her mother were/are concerned about the baby being around the dog, but I never leave the two of them alone. I mean, Shadow is a DOG and you never know. But he has never shown aggression to us or the baby. Just the birds and squirrels in the back yard.
Best of luck to you! Welcome to the journey!