I don't really remember "learning" to trackstand... but my "good" side is with my left foot forward and the wheel turned to the left. I can also do it with my right foot forward, although on some of my bikes I have to be careful of toe overlap. I can take one hand off the bars either way, but I can only take both hands off if my left foot is forward because I do it by standing up and sort of bracing the top tube against my knee. With the right foot forward, the wheel is turned away from the forward knee, so there's no good way to brace with no hands.
I can sit down either way, but it's harder than standing. If I sit down I have to actually think about it, whereas when standing it's just a habit and I do it without thinking at all. I rarely put a foot down at stop lights.
But I haven't learned to do it with the wheel turned to the right, and I haven't really tried to practice it that way either. Actually it's not really something I have ever practiced, it's just a habit.

I'd kinda like to try the kind where you put a foot on the front wheel, but I have fenders on all my bikes so that doesn't really work.

As to whether it's useful, I think it is occasionally but certainly not critical. The best practical reason I can think of is that a lot of times I will trackstand with my left arm sticking out when I want to make a left, and then when there's an opening I can make use of it faster. The other thing is, if I'm standing there in the middle of the street with my arm out, drivers will ignore me; but if I am trackstanding (even if I am perfectly stable that way for as long as it takes), they're more likely to let me make my left turn because they think I'm halfway through making it already.

It's also useful for those occasions when from time to time I ride with one hand and carry something in the other, like a cup of coffee or takeout or something awkwardly shaped that I'm only going a short distance with. With one hand occupied, it's easier to start and stop (especially at lights) in a trackstand than to put a foot down. Before you ask, I ALWAYS have two brakes on my fixed gear, and if I'm carrying something, I carry it in my right hand so I can use the front brake with my left. I still signal left turns even if I am carrying something in my right hand.