Dog Potty Training / Behavior Issues

Get him (and his poo) checked out by the vet. Strong smelling poo, especially if its runny, can be a sign of disease such as parvo (very serious) or parasites such as coccidia (not as bad, but still needs treatment)

If one or more dogs are vomiting and being incontinent in the household, I would be very concerned you have illness going on. Hopefully it's just something they got into, but better safe than sorry!

Once you get the health thing sorted out, then tackling the housetraining comes next, and perhaps a lot easier. first off, I just want to say dogs are NOT "stubborn" or "refusing house training".... they just don't have motivations like that.... that being a sort of human emotion we are putting on them. What is most likely is that pup is "trained" to go in his crate.

Somehow in his early experience he learned this is THE place to go... the only safe place. You are going to need to be very creative in figuring out a strategy to fix this pattern. What you need to do is manage him in such a way that he ends up pooping outside with you there so you can reward him for going in the right place. (use a super tasty treat!) You must never scold him for pooping or peeing in your presence, even in the house, because he won't realize the reprimand is for the location... he will think it is "dangerous" to go potty NEAR a human. Usually it is such reprimands that are the origin of these difficult to sort out house training problems.

So you might try having him tethered to you on a 6 foot leash as you move around the house. Take him outside periodically and praise/treat him if he pees. Keep doing this. Eventually he is going to have to poop. The trick is make sure he cannot poop in his crate, but rather poops outside. Therefore, don't put him in his crate. Keep him tethered to you and keep taking him out every half hour or so. If he starts to poop in the house while with you, just stay calm but interrupt him with an upbeat "Let's go outside!" (or what ever phrase you use for going out) and whisk him outside. Maybe he will finish pooping outside, maybe he will be too worried about pooping near you. But the idea is you need to set up this first success somehow, even if that means having your dog tied to you for 2 days. When he finally does poop outside, after he finishes, have a party! Tell him what a great dog he is and give him a jackpot of treats (keep them with you at all times so he gets that reward right away)

Once you have one success, the second one should come a little easier, then it is a matter of building up the positive experiences. Clearly you cannot use his crate during this time.

Again the principles are:
1) prevent the possibility of him pooping in the wrong place by not allowing him access to his crate and keeping him on a tether to you so he cannot sneak off to a back room to poop
2) REWARD his successes

3) Most important, eliminate any punishments for mistakes. Dogs rarely learn what we want them to learn when we punish them for house training accidents. Instead of learning the "inside/outside" distinction what they learn is "safe/dangerous" distinction. That is it is safe to potty while owner is not present and dangerous to potty if owner is around. Thus the dog becomes afraid to potty with you nearby, so you can not get them to go out on a leash and reward them for going in the right spot.