First, thank you for rescuing your boy.
Do you know anything about his past history? If the shelter were about to phts him, does make me think he's got a few 'issues' which need gentle, patient and understanding to resolve.
He's now found someone who loves him and doesn't want to let you out of his sight. So keep that in mind and be very patient with him, it's great he's grand with your other dog, that will help. Also loads of toys, Kong filled treats when you have to leave him alone work wonders.
Blacknym and Southern Girl have both given you excellent advice. And yes, the pooping is more than likely caused by fear, or he just doesn't know it's inappropriate maybe because he wasn't taught the correct behavior in the first place. I had that same problem with a rescue girl a while back.
Basically what I did was train her as if she were a puppy. Take him outside when he's eaten, if he looks 'interested' i.e. sniffing around etc., to a chosen spot wait with him until he 'goes' and then praise/click/treat like mad. He'll catch on, but it does mean putting in some work before that, just as you would with a puppy.
With his crate Blacknym and Southerngirl have covered that really well, so all I will add is maybe once he's calm as per above post/videos also leave him with a puzzle toy filled with high value treats just as an 'extra' bonus, plus it will keep him occupied and mentally stimulated (which is more tiring than exercise) when he's crated. Also, make sure he doesn't need to 'go' and is alerting you to that fact. Even if it's the middle of the night, you may need to check him (it's a pain and sleep deprivation isn't the best) as he may be trying to be a good boy and alert you to the fact he needs to go. Even with older dogs, especially ones with a possibly traumatic history, sometimes get 'caught short'
With his snapping, don't 'react' to that, it's easy to do so, just remain calm, turn away (warn people to give him his 'space' too I'm sure you're aware of the yellow leads/collars and bandanas for dogs who need their space) and ask for sit. When he's sitting nicely, then click/treat/praise. Also, don't try to 'make friends' with him, allow him to come to you. You can start just throwing treats, casually, as you pass him not asking for anything at all. It does sound as if he's afraid of people. Throwing treats casually, so they land just near him not on him of course, is something visitors can do when they visit. It may take a wee bit of time but he will eventually realize that those big scary humans are actually very nice people who love dogs
Oh yes, keep low and don't look directly at him, just glance, look away, lick your lips (a calming signal to a dog) yawn (loudly) or, this may sound weird but it works, if outside pretend to nibble the grass. All these things are calming signals to a dog and you do have to learn 'dog speak' to communicate to a nervous dog you're not a threat. You may feel and look a bit odd but only to humans, not to a dog!
Taking on a rescue dog is wonderfully rewarding, keep that in mind, but it does take patience, tons of love and loads of understanding. We'll all help you here, a lot of us have been through it all before and can well understand your frustration, so don't beat up on yourself, your human frustration is normal, happens to us all. Three deep breaths really does help
The more relaxed you are, and that doesn't mean just 'accepting' in appropriate behavior, the more relaxed your new boy will be in his new surroundings. Post again if you need more help and we'll get you through the initial 'settling in' period.
