There is a lot of info about submissive urination on the web but I've included one link below.

She's still trying to adjust to her new household so if you take everything slow and follow the suggestions in the link, you should see improvement.
For traveling in the car, I would keep her in a small crate until she's no longer urinating and as far as her hating to go out when it's cold or wet...well......that's pretty typical. You may have to just pick her up and carry her out, then spend time outside with her to make sure she's doing her business. Do get her a coat/sweater for the winter since Chi's appreciate them.

Go slow with the training and don't expect an overnight fix. It will take time and patience.

Here is a link that might be useful: Urination

If you can crate her while you're gone, you can carry the crate out and open the door and let her out. The idea is to make as little fuss as possible so she stays calm until she can relieve herself, so don't talk to her or anything until she's outside. If it isn't possible to crate her, just try to be as low-key as you can until you get her outside.

I agree with annz about needing to stay outside and make sure she goes. I have two small dogs and they need someone to go with them and remind them why they're out there or they'll often turn around and come back in without doing anything. Also, a lot of female dogs seem to not like to potty in a strange place, so she may need more time to get used to your yard and make it her own. Maybe she's holding it in when you take her somewhere, too, and then she has her accident when you lift her to take her home.

Another possibility is she could have physical urinary incontinence issues rather than submissive peeing. It's fairly common in spayed females, so you may want to ask the vet.

And be sure to take her outside often until she gets used to everything. Don't make her wait as long as your other dog. She may need to go more often.

Thank you all for the useful information. I am going very slow with her so that she understands this is her home and no one is going to hurt her. She's very loving and playful, and uses pee pads. I'm grateful for that and I will speak to our vet about possible incontinence. Thank you all again for your suggestions. Oh, I almost forgot. She has three sweaters in different styles and she's quite the Houdini at taking them off, along with her collar. Loud and clear she doesn't like them!

I once had a pup that had submissive urination, exceited by greeting and just leaked. One thing I was told and it helped was "don't tower over the dog". Scooch down to their level when you greet them. Also stay calm and immediately bring them outside. Eventually it stopped.

I was going to suggest not repeating the actions that you know will cause this. I like deb18's idea about crating her so she remains calm when you come home. If you don't feel like crating her then try ignoring her when you come in so she doesn't get so excited. Do you greet her by paying her attention and leaning over her to pet her or even pick her up? It seems like she may have an issue with being picked up. Maybe it scares her to have someone lean over her and grab her. Maybe try not doing that for a bit to see if it helps.