How can we help our high anxiety (neurotic) dog?

EmmO Asked: How can we help our high anxiety (neurotic) dog?

I'll explain first, so that maybe I could help shine some light onto the situation and to clarify exactly what it is we're dealing with here..
We have a1 1/2 year old malti-poo (or 'moodle') named Jack whom we got from my aunt who saved him as a puppy from a 'back yard breeder' back in Oklahoma. He's a fantastic little dog, great with the children, great with cuddling.. and wants nothing more than to be loved loved loved, all the time. Just the sweetest thing!
However, he is extremely neurotic.
Examples:
-If put his leash on him to take him to go potty, (we live in an apartment complex), or simply to take him for a walk, he pees as soon as he sees it and spins around in circles while doing so.
-Same happens when I try to put his collar on after a bath, he fidgets because he doesn't like being held still, then pees.
-If I pick him up, especially for bathtime, he pees.
-If he remotely thinks there might be a chance that my husband and I are mad at him when we go to pet him (he constantly thinks this, it seems, but we're rarely ever mad at him!), he pees. It even happens sometimes when anyone comes close to him to pet him, pick him up.. things of that nature. I even get nervous when I tell him to get off the couch -in the sweetest 'stern' voice you've ever heard in your life, because he'll pee on his way down and then walk around the couch, still peeing- or when I pick up his toy that he drops next to me when he wants to play.
The point, really, is that he's very nervous and anxious, and pees about it.
He also poops in our room and our daughter's room.. but mostly her room. The weird thing is, though, he IS potty trained! He tells us when he has to go, and we take him outside and poops and pees just fine! (Even after he pees inside on the way TO go potty!) We can't have him pooping in the house, ESPECIALLY our daughter's room mainly because she has an immune deficiency, ('hypogammaglubulinemia unspecified with low b-cells'), and can't exactly be exposed to animal feces on and around her toys or clothes.
This is getting pretty lengthy, so I'll skip any more specifics (unless you ask them!).

The point is, we're at a loss. We're frustrated as hell, because if he continues this behavior, we'll have to get rid of him for my daughter's health (and life, really).. which would break our hearts COMPLETELY, but will be what must be done. We want to help him, so as to avoid doing so, but we really have no idea where to even start with him. I looked into it and found that his sister was born with no knees on her back legs, which clearly leads me to believe he's been terribly inbred (or over-bred?).

We love him, and we want what's best for him, but we don't want to have to give him up (or give up on him!). What can we do to help our poor Jacky get over his anxieties? Please help us!