I go with the sunroom, too, and here's why. You can build a shed for cheap at some point, and until then you'll probably keep one car inthe driveway, and all the bikes and basketballs in the garage along with the yard and snow tools. The sunroom will make a great playroom while the little ones are little. I've seen that sunroom with nothing in it but a low-pile play rug and Lego. Or balls. Or plastic hockey sticks. Or those big Little Tykes (slide, trikes, trains) Or Hot Wheels. Or dollhouses. Or art projects (easy-to-clean tile floor). Usually just one or two of those. But a great playspace, near you but not, outdoors but not. After they are done with those types of toys, it can be dining space and added living space. You don't need the shed until Boy #1 gets a car. Oops, didn't mean to scare ya lol! Upstairs, I wonder if you can steal some space from the game room to make a reach-in closet for Bedroom 2, and use its walk-in closet to expand the bathroom. 5 boys will need more counterspace, more drawer storage, and (I can't believe I'm saying this, I'm usually a 1-sink gal) 2 sinks. Downstairs, consider eliminating the study closet and adding a shower to the powder room.

What will you use the space for? Does it have to stay gray? To me a sunroom is bright and cheerful

Our floors were open to ground also. We had foam sprayed in from the underside. In some places, where it was a tall enough crawl, we had pink stuff, foil covered foam installed under the floor joists, taped and touch in foamed along edges. We did get occasional leakage between boards, once peeled out, ya don't even notice.

White beadboard nailed directly to the joists, no coffer. If bottom of joists are significantly unaligned, you may need to install strapping first. This will result in the boards running in a different direction.