As long as you ruled out any health problems, she's just extremely submisive. Accept it, deal with it. Doesn't mean you did any thing bad to her, doesn't mean she thinks you're a bad or scary person. In fact, to a point, you can see it as a compliment as she obviously sees you as her leader. Don't let this make you feel guilty or bad about yourself. Just keep in mind she is a very sensitive dog so keep any corrections mild and even avoid them if possible.
If she's being slinky (that's what we always call our dogs when they are acting submisively, the typical low to the ground looking up like they just did something bad) and cowardly towards you for no obvious reason, again, don't take it personally. Don't come up with silly ideas that she's mad at you, or you did something yesterday to hurt her feelings.
My last dog urinated submisively like that too. Almost always and guaranteed when I saw him first thing in the morning after waking up, or when I came home from being out for a while. I couldn't even look directly at him or he would loose it. Even if I stopped and stood there (and he was anticipating some sort of contact from me, eye contact, petting, etc) he would loose it. So the routine was always when getting up in the am or comming into the house, do not look at the dog! Walk quickly, call his name plesantly, say hi plesantly, walk him right to the back door, open the door, and say "go potty". He would then run out, do a pee, come back in, and it was safe to greet him. He actually wouldn't show extreme submisive behaviour once he came back from his pee. He would look happy and relieved and just wanted to do normal doggy greeting protocols. I think his previous owners used to punish him for the submisive peeing, and he knew that was wrong, he just couldn't help himself.
Also, similar as to what you said about your dog, he was fine with strangers. He was very friendly and outgoing, just looooved people. As soon as he got to know a person for an extended period of time (a few visits perhaps) he would start doing the submisive pee to them.
The best thing you can do for this dog is to try to build up her confidence. Avoid corrections if possible and if you do have to correct (like with the cats) make sure your correction is to a minimum. Meaning the least amount of "force" it takes to get the message through. Perhaps you don't have to yell "no", maybe if you say it directly to her in a low tone growly voice "Bingo, nooo" that will be enough and won't send her off into the slinky mode. Plus for the cat thing I would add a second behaviour. I would do something like "Bingo, nooo", then in a chearful voice "come" "good girl!" and then some sort of sit, down, shake a paw, what ever routine. I would like to follow the negative input to her with some direction and a positive.