hardwood versus carpeting?

hardwood under all the carpets
All the floors upstairs and down in our new house have hardwood floors. We are trying to decide if we want to clean up those hardwood floors and use area rugs or put down new wall-to-wall carpeting (or some combination of the two).

The soon-to-be den will have some walls taken out of it any day now, so we'll need to either patch them with regular subfloor or matching hardwood. The upstairs bedrooms we have already decided to carpet and will just put subflooring up there to match heights where the dormers were widened.

So if the upstairs is carpeted, what are your thoughts on the downstairs? The office and the sewing room would be nice to not have thick carpet (which is not friendly to either rolling chairs or finding sewing needles). But then there's still the living room, dining room and den.

What are your preferences for flooring, and more importantly, why? We have five pets, one of which (Sarah) creates tumbleweeds of hair all over the house. Sarah also can't climb the stairs anymore and doesn't have the surest footing in the world, so we would definitely need lots of area rugs if we had hardwood floors downstairs.