I will attempt to answer some of these for you but there are a few guys that can probably do a much better job. Hopefully they will come on and post answers with rule specific language.
1) Stealth and LOS for that matter were troublesome rules for me to follow as well and I think give most new players a headache to understand. Stealth states that the piece can not be the target of an attack unless he is the closest and within 6. So if the fig is outside of 6 squares they can not be targeted. However, like most things in this game there is a counter
Someone with accurate shot can effectively counter stealth and target you from any distance and no matter if you are closest or not.
*as a note for superstealth there is a "bug" that if the fig is within 6 as per the stealth rules they are considered the legal target as the closest but since SS does not allow anything but an adjacent attack that shooter can not attack anyone with a normal ranged attack. I guess the rationalization is he is trying to figure out if he saw something there or not
2) This part I still get confused on myself. Basically I think this errata was thrown in there for the screwups on cards. It used to be that only if the name was stated in the skills did a fig become part of a group such as Black Sun. There was an errata that allowed a fig to be considered Black Sun if they contained it in their name or in their abilities. The same goes for most other name related abilities I do believe.
3) Greater Mobile allows you to move up to your speed and still make all your attacks ie double, triple etc.. while mobile only allows you to make one attack. Your speed is usually 6 unless there is another speed printed in the abilities list. You can move part before the attack(s) and the rest after. You could also move all before or after, although that would be a normal move for a mobile attack
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I hope that helps and I didn't confuse you more.