Yes, it is most likely a saddlebred. Saddlebreds are terrific horses. They are proud, showy (even if they are just pleasure horses, they think they are hot stuff). I am a saddlebred trainer and owner. I teach saddleseat and I teach hunter. here are the answers to your questionss:
1.) The saddle: the saddle type really depends on what the horse looks like. Normally, with a saddlebred you ride saddleseat and you use a close-contact saddleseat saddle that is cut back in front. This is because saddlebreds set their heads very high. Because of this, the line the neck makes in front of you is up, not straight out and not down (like a quarter horse might be). This means that some horses need a little space their for their neck to comfortably move. You do not HAVE to ride saddleseat on a saddlebred (although I am a bit of a purist in that regard.... 
She could ride hunter if she wanted to. The issue with hunter is again, going to depend on the horse's head set. But, if this horse is strictly for pleasure, then it won;t matter that she is riding hunter on a saddlebred with a higher head. Plus, the horse is 20 so he might not set his head as high as he used to. he also has not been used much and was a trail horse, so my guess is that he may not have been trained to set his head to begin with. All of those things could be positives in terms of your friend being able to do hunter with this horse. So, back to the saddle issue, if the horse's head is not incredibly high, a hunter saddle might work find.
2.) As for saddlebreds in general, they are amazing horses. They were created to be plantation and parade horses. They are very aware of their surroundings and can be very brave. They are loyal and funny. Mares can be very stand offish, but a gelding is great. I am not a big mare person to being with, but saddlebred geldings have a close place by my heart.
The gates are smooth. the only issue with riding huntseat in terms of the gaits may be the trot. The trot is so fabulous on a saddlebred. But, in a trained saddlebred you post MUCH higher than you do in hunter. You must because the trot is a great deal "bigger" However, this horse is a trail horse and older, so again, I think that she should be fine. Working on slowing the trot is very easy and even big trotters can be used for smaller trots. I have a 7 year old gelding right now that has a big trot but I use him in kids lessons with teenageers and he can be slower and allow them to post in a more hunter-traditional style. He can be ridden by hunter riders as well.
3.) As for the "5 gaits" as you say. There are essentially 4 types of saddlebreds I will be very simplistic here because it can get a bit complated). The first kind is the "country pleasure" horse (these are the show divisions that you would see at a real saddlebred show). The country pleasure division is for horses that are more natural, they do not step as high, they do not hold their head as high, they do not have fancy shoes on, etc. They do the walk trot and canter. Then there is "show pleasure" whcih is a bit fancier, the horses step higher and may have a higher headset. Then there is "3 gaited" these horses have a shaved mane to accent their extreme headset, the neck is set high and arched beautifully because the nose is tucked in towards the chest. This horse also has a tail that has been set to go UP and then down, in an arc. These horses wear tail sets all the time to keep the tail correct (it is pretty awful, I no longer allow tail setting with my horses). Then there is "5 gaited" This horse is probably an old 5 gaited horse who was taught the two extra gaits originally and then never showed in his older years. The five gaited saddlbred does the walk, trot, and canter and then the two man-made gaites: the slowgait and the rack. The first is (for the most simple explanation) a slower version of the second. The issue is that these gaits, in most horses require some very specific signaling on the part of the rider.
many horses are taught that a slight and alternating movement of the hands (back and forward about 1-2 inches, alternating hands, and doing so very quickly) matched with alternating taps with either foot. The gait feels TOTALLY different than anythinig you have experienced. It, on some horses, has a slight side to side feeling but is often VERY smooth.
It would be best to stick to walk trot canter for the time being and wait to try the other two until the horse and rider have a set communication system that works for both of them. You never know what signal the horse was taught initially so you don't wan to frustrate things by asking for it incorrectly too soon.