A bathroom is probably the best option for the brooder while indoors, simply because it's made to be cleaned. Past that, maybe a laundry room? If you need to put it in a carpeted room I'd personally think about putting down a tarp or a lot of cut up cardboard first.
As far as bears, probably...
The cardboard box is way too small right off the bat.
The "brooder box" should be ok up to about 2 weeks (others will say 4 weeks = 1 sq ft per, but by 2 weeks they'll be bouncing off the walls). Make sure it has a secure cover/top/lid, and good ventilation.
Just for reference my brooder is...