Not silly questions! You can't be expected to just know all of these things.
1) A running martingale, in my limited experience (I have never used one, just seen/heard of them) is just one step up from a standing one. Same idea - keeping the horse's head from getting too high, I think the difference is that the standing attaches to the noseband and the running attaches to the reins. In the running, the rings will apply pressure on the reins if the horse's head goes above where the rider has control. I think it's mainly used by eventers.
2) I don't know much about bits, but from what you're describing, could it be a slow twist full cheek? Similar to this? Slow Twist Full Cheek Snaffle Bit | Dover Saddlery
What Saskia said about this is good advice.
3) That is the way I learned to cue for a canter. Giving support with the inside rein helps them balance for the inside lead (which is what you're cuing for with the outside leg back).
4) Work on your jumping position a lot on the flat to strengthen those muscles. I have just started jumping so I'm afraid I won't be much help on the jumping questions.
5) Like Saskia said, you might be straight. Have someone watch you ride and ask them what they think.
6) I would work more on your position and holding your balance over jumps before you mess around with leads.
7) You probably have a loose grip on the reins to allow your horse to move. The 'forgiving' part of your arms is your elbows - your hands shouldn't move. I struggle with this A LOT. A little exercise for you: my instructor will pluck two tail hairs from my mount and make me hold one in each hand at the beginning of the lesson, with the objective being to still have them by the end. Gloves are not necessary.
8) I've never heard of doing that before, but it does sound like it makes sense. Less stuff in the way when you're trying to get the bit in. I may just have to start doing that now!
9) Some horses are taught to automatically open when the bit is placed under their mouth. Others you have to put your thumb in - some were never taught, some have become sour after years of having the bit shoved in or clanked against their teeth, etc. As long as you make sure that they are in no discomfort when you put the bit in, using your thumb to get them to open is fine.
10) I don't know how long polo wraps have been around, and as far as support/protection I find them virtually useless. A lot of people just like the look of them. There is a learning curve to putting them on - there's a certain direction they have to be rolled onto each leg, and they take a lot of practice to get the right tightness and eveness. Too much work for something that does so little good.
11) I wonder if their saddles fit them correctly? Or have had saddles that didn't?(TBs are usually hard to fit).
12) Sounds like you're describing a flash noseband. It helps keep the horse's mouth shut.
Good to hear you enjoy riding! Keep asking questions, especially on this forum - everyone here loves to give their opinion