Well first of all, you sound like you are taking poker seriously, even playing at relitavely low stakes, which is a good thing, that gives you an edge over a lot of players down there straight away.
I will answer your questions from my own experiences.
(1) You got it right, the lower the level, the more people are to keen to shove all their chips into a dry pot, without actually thinking about it, very few players will fold when they have some money in the pot, and they over value Ace Jack and lower. There are some good players at $10 level, really there are, but there are a lot of awful players too, the key is to pick the bad players out.
If you can beat the $5 sng, you can almost certainly beat the $10 ones, and more than likely pay a more fair amount of rake. Stick to a simple strategy, what ever is working for you now will work for you at $10. I am not that certain it gets much better at $20 level, there are bad players at all levels, I have seen some right donks on $200 sng, but the percentage of good player is much higher obviously.
Probably need to work on your flop play a bit as you move up, know how to fold top pair top kicker,and how to pick the right spots to make your moves.
(2)To be honest I don't think there is any amount of games that you need to move up, just sufficent bank roll. I have heard some say 20 buy ins is enough. But I think 50 is a much safer amount, then it gives you chance to move back down without to much damage being done if things are going bad. That is the main thing, if you aren't beating the game, move back down, and work some more on your game. Don't panic if you start off poorly, but don't play scared, if you are worried about your bank roll, move back down right away,
(3) The money is in first place in 6 seat sng, so you need to get a fair share of those in as well, it will help you get your ROI up no end. Finishing second often isn't a bad thing. Work on your heads up play, sounds to me like you play a tight game, and scrape in to the money, which is no bad thing, In my experience going in to the heads up short stacked, just be aggressive with any decent cards, they will often call you with nothing just to try and finish you off quickly. Only takes one double up and you are right back in it.
Hope that has helped in some way