depends on if you're using mods or not as far as hull integ goes.
i know that in stock sh3, you can see it by going to the crew assignment screen. should be in the upper right some where.
if you're using gwx version of sh3, you dont see hull integ.
anything damage wise you would see from the crew management screen. including flooding. not to mention your crew will be more than willing to tell you if there's flooding, and if all else fails you'll notice your boat sinking into the depths.
i know when you go back to base, your hull integ is fixed. i'm not sure about systems though, because i've always fixed those while at sea. if they're destroyed, they are repaired at base before your next mission.
