Yes, the oil pressure light. It should go on when you turn the ignition, but switch off when the bike's running. When it stalls it'll probably turn on again.
let me go through your questions:
1. If your idle screw wasn't set right, it wouldn't idle well in neutral either, so that's fine.
2. Yes, but if it was you would feel the bike try to take off and stall. I'm guessing this is not the case. But if you want to be sure, try walking with the bike when it's not running. In first gear you should have a hard time, but with the clutch in you should be ok.
3. No, they don't.
4. as Lou says
5. The float bowls are part of your carbs. So yes, they have something to do with it, meaning you need to go in them to clean. There could be gunk in there, and it's likely there is.
It might seem like it has something to do with the gearbox since it stalls when you put it in 1st. But I had this as well, and the carbs were the problem. When you put the bike in 1st, it always shakes a bit. This little shake is enough to make the bike stall, if it's nearly stalling already. So yea, basically what we're saying is, clean your carbs.
Ultrasonic is nice, if you don't mind paying for it. You'll still need to probe through the passages with fine wire, but it'll reduce the chance of them getting clogged again. Just cleaning the carbs yourself with some carb cleaner, boiling them in vinegar or lemon juice, works as well.