Moisture from under the floor - is this bad?
I don't believe that's rain, just condensation. An uninsulated concrete pad whicks up moisture. Can't be stopped without alot of work.

Your best bet is to surround the outside perimeter with styrofoam insulation board, the thicker the better, up to 2". Then you can try sealing it as you stated. Make sure to leave a couple of those holes open for ventilation, or it'll get sweaty under there.

Back to the pad condensation, a good "college try" could be to dig down a foot or two vertically around the pad and insert 2" foam vertically, flush to ground level, the backfill. That won't stop condensation, but it will reduce it by breaking the frostline, which causes extreme temperature variations between the pad and the ground, and thus condensates.

Like I said, alot of work.