First, if you lunge her, don't lunge her in the tight circles you are used to with a trotting horse. Gaited horses shoulders have a bigger, more sweeping motion and need more room
Second, she is beautiful! It's really hard to say from that video but she almost looks a bit off on the left front; it could only mean she might have stepped down on a stone coming around the turn.
She also appears to move her head side-to-side; again hard to tell. If that is the case, she will rack or step pace; neither of which are bad as far as I'm concerned but many don't agree with that and try to force these natural rackers to perform a running walk.
I have two that do the running walk and one that step paces. The step pacer is every bit as smooth as the other two.
Bits: Use whatever SHE is comfortable with - even bitless if she's ecstatic with that.
Two of my Walkers wear very low port curbs with swivel shanks (I prefer swivel shanks), the third has always worn a hackamore.
The TWH I lost in a freak pasture accident wore a Dr. Cook's Bitless and he loved loved loved that bridle.
Saddle: I honestly would not spend money on a professional fitter until she gets back in condition as I promise you, just like any horse, the saddle won't fit in six months or a year.
Walking Horses can wear any type of saddle that fits them properly, including saddles made for other breeds. The TWH that I lost wore my Arab's saddle, until he filled out, because he was short-backed. I then bought him a wide AmTech synthetic endurance saddle.
The key thing to saddle fitting with a gaited horse is to allow the shoulders plenty of freedom of movement for those huge sweeping motions they make. And, if the horse is short-backed, they are not happy if the the saddle is too long.
At any rate, I would wait a bit before buying her a custom saddle:)
Do you have current still shots of her from the side and even one showing a clear view of her back?
As others have said, basic training is basic training<--- they all whinny the same
Everyone has a different opinion but the way I was taught years ago, was to keep my hands low, the reins a little loose and sit back on my pockets when I wanted my horse to gait.
Keeping them in their gait is a little like power braking your car; pulling back on the reins slightly while lightly nudging them ahead with your lower legs at the same time. Hope that made sense.
Unless you're planning to show, getting the gist of riding her is not as big a deal as some people like to make it. Don't over analyze once you figure out what her cues are; just go trail riding and enjoy smelling the roses at something a lot smoother than a trot
While she might be rusty from not being ridden if she insists on trotting, have a chiropractor look at her and have her hoof length checked. As long as she has papers to prove she is 110% gaited, there is no such thing as a gaited horse that won't gait. They can get ruined right into trotting all the time or they can have something wrong with them.
One thing I have noticed with my Walking Horses is that when something structural is wrong with them, the first thing they do is trot a lot. Unless they're behind my Arab in the pasture and they will mimic him.
One of my Walkers will trot in the pasture when his back toes get too long.
The one I lost in the accident suddenly couldn't hold his gait under saddle and would rather trot. His Atlas bone and sacrum were both out of place. Once the Chiro worked on him and he got some rest time, he was a gaitin' fool/
Hope this helps some