MyBoySi, in my experience, the longer you can keep a horse going with a rider, the longer their life will be. Horses are about like people in that respect, so long as they can work and do
work, they stay in shape and they stay healthier. However, once they are retired and stop working, they lose condition quickly and their overall health goes downhill in a hurry.
OP, I suspect that you're not doing anything wrong to cause
the behaviors. He's an old horse and has probably had those habits for years, you are just unsure how to correct them due to your inexperience.
One thing you really need to learn, not just for this horse but for any horse in the future too, is to hop along if they move when you've already got your foot in the stirrup. Ideally, for mounting, you should have your reins in your left hand and that hand on the mane. Your right hand should be on the saddle horn. That way, if the horse begins to move or walk off when you're in that precarious position, you can pick up your left hand to control them and still be secure in your position.
When he moves as you walk up to his side or when you put your foot in the stirrup, I would put him to work. Make him lunge at a trot in little small circles around you. Or, you can back him up fast for a long way until he's wanting to stop. Then, when you let him stop, immediately go back to his side and start to mount. If he moves again, then put him to work again. Wash, rinse, repeat until he stands stock still for you to walk up and mount. If he does stand well, then let him stand for a couple of minutes and give him a scratch or two on the neck before continuing.
The catching thing, well, that's something that most horses do if they are actually worked for a living. Most of my horses are that way, but I've learned how to keep it from getting too bad. Sometimes, instead of just catching them to ride, I'll catch them and not ride at all. After I catch them, I'll spend time grooming all their favorite spots, letting them graze on fresh green grass, or giving them a few handfuls of their favorite feed in a bucket before turning them back out. That way, they don't associate being caught with having to work.
With the problem you're having getting him to lead well out of the stall, he's just being stubborn about having to work. You're not doing anything wrong by turning him in a circle to get him to move on. That is exactly how most folks would tell you to deal with it because it "unsticks" his feet and gets him moving again. I've found that using a butt-rope is a little easier and more effective though. When I have this situation (usually on fairly young horses), I'll take a long piece of the heavy orange baling twine and put a loop in one end of it. Then, I put the loop over the horse's hindquarters like this
Then, as you're leading him, if he balks, just give a gentle bump with that rope. Some horses will jump forward so you need to be prepared for that, but so long as he's moving forward, just go with him and don't get after him for jumping. I've found that it only takes a few times of this before they stop balking completely and you can do away with the rope.
As for the gate-sour problem, that's not uncommon for former performance horses like him. They are taken into the arena where they work hard for a couple of minutes, then they are taken out and the cinches are loosened and they get to relax tied to a fence or a trailer so they associate leaving
the arena with being comfortable.
Do you ever ride outside the arena on trails or anything? If not, you might start. Take him out for a long trail ride or school him in a flat place near but outside the arena gate then, when he's hot and tired, bring him home and take him into the arena. Take him to the middle, facing away from the gate, get off, unsaddle, and turn him loose to roll and relax inside
With him being as old as he is, that may or may not fix
the issue, but it should help to minimize it.