My first thought would be to do the fecal (giardia and coccidia would be my primary concerns) and also cut back on his food. If he's eating and eating and eating and not gaining any weight, then either is a naturally lean dog, or something is medically wrong with him (like worms). Either way, feeding him more isn't going to help. If you need to try to keep weight on him, feed him a nutrient/caloric dense food and/or supplement with satin balls (
http://www.holisticdog.org/Nutrition/Satinballs/satinballs.html).
Kiba, the Pittie I fostered, was a naturally lean dog. He stayed at the same weight (borderline lean/skinny) eating four cups a day or eight cups a day. Didn't matter. But when he ate eight cups a day, he was pooping out of his ears. LOL His new family had him neutered and I'm assuming that cut his metabolism down as the picture updates the send me shows him with a normal amount of fat over his ribs, which was impossible before.
I'd crack down on re-house breaking him. If he has no den instinct, then it will be difficult. Cooper was never properly housebroken - he wouldn't go in the house, but that was because he had an owner equivalent to a doggy door. If he's left loose in the house over night, he'll crap on the floor. Luckily we don't have that problem when he is in his crate as he at the very least considers his crate his den, but we haven't had much luck teaching him to expand that concept to the rest of the house.
And if it only happens at night, barring any medical problems, I personally wouldn't be opposed to some kind of indoor/outdoor dog run for him. He can still be a house dog - no need to banish him outside and only outside - but just use the run as you would the crate. That's what we do with Rose. No fault of her own, but she's incontinent and the medications were hit and miss. When she's let out during the day she's fine, but over night and while we were gone long hours she'd leave a puddle and it was ruining the floor. So now she stays in her outdoor kennel as her "crate" and everybody's happy.
Even if it was a temporary solution while you worked with him on it, it would be worth it if it saved your sanity.