Sounds like you're doing it right. Here's some feedback:
If I like the horse, I am having a trial period of 2 weeks with the horse before it is officially mine.
That's a good sign if they're willing to let you do a trial period but typically most people won't allow it
During this time I am going to get a vet check by my vet and get the previous vet records.
Most states don't allow you to get a copy of vet records so double check that first. The current owner would have to and I don't think that's a normal thing to ask the owner for unless they're making a claim about some medical procedure like vaccines. Good job on going for a vet check. Depending on how much you're paying for the horse it may be worthwhile to get blood drawn for a toxicology report. It's expensive so don't bother if you're dealing with a horse under 10k. The general tests are usually good enough.
My trainer is coming with me both times I go look at the horse.
Is it rude/weird to ask the owner if I can watch her bring the horse in/out of the field before and after we ride and watch her tack and untack the horse? How the horse is on the ground is also going to be a big factor for me.
Absolutely normal to ask to see that and you should
Could I also ask to take the horse out on the trail? How would that work? Would the owner go with me on another horse and we would be out there for a very short period of time - just so I can kind of see how she is?
You can ask but not every owner is willing to do that. A lot of it depends on the price of the horse and how much effort the owner wants to put into the sale. If there's a trail accessible from where the horse is it's more likely they'd agree to let your trainer ride it on a trail. When I bought my horse the owner agreed to ride it on a local trail with me walking with them.
How long do you think I am expected to be there? How long will I be riding for? I dont want to take up too much of her time, but I am also very interested in this horse and dont want to feel rushed or that things I am curious about have gone unanswered, especially since the horse is so far away - I dont want it to be a wasted trip.
Just don't waste the owners time. If you're starting to feel like it's not the right horse just let them know. With such a long distance drive spend at least a couple hours there the first time. Help groom and tack up the horse. Let the owner ride the horse for you, let the trainer ride, and then you ride (if you're comfortable). Work in a short trial trail ride if they allow you.
Since the owner hasn't sold the horse in a couple of months there may be something wrong with it or they're just not motivated to sell it. You'll need to look for any signs of them lying or being dishonest (so many horse people are). Do not pay a deposit on a horse that they've been trying to sell for months, they're obviously not in a rush. If you really think it's the perfect horse, you can sign a contract for a conditional sale if the vet check comes back clean. That will give the owner some assurance that you're a serious buyer.
Also, what should I wear? is there anything special I need to bring? I was thinking breeches, tall boots, and either a polo or tshirt. and I am bringing my helmet. Also, when I go to pick her up for the trail period/hopefully to keep her, am I expected to bring a halter and lead rope for her? and what about blankets? Its very cold here, and she doesnt have a lot of hair, would they give her to me with all this stuff and if I end up keeping her, just ship the stuff back to them?
Bring your own stuff. Everyone has extra old rope halters and ropes around they're probably be willing to give away but you should still bring your own. The other stuff is your responsibility. If the horse was blanketed before you should get a blanket. You could also ask your trainer to borrow some until you get your own.