Invisibility is always at 100%. I believe you are thinking of Chameleon, which unlike Invisibility, remains in effect when you attack someone, pick their pocket, go through a door, etc... You can find all about it http://www.uesp.net/wiki/Oblivion:Chameleon.
To trap souls, you need an empty soul gem that is large enough to hold the soul you are trying to trap. Then you cast Soul Trap on your target, and as long as they die while it is in effect, their soul is svcked away into your gem. Azura's Star is of course a Grand soul gem, and is reusable so it is your best option. There is lots more on Soul Trapping http://www.uesp.net/wiki/Oblivion:Soul_Trap.
Soul gems (black or not) are used for recharging magic items. When you click on a filled soul gem, it will show you a list of all your magic items that are not fully charged. You just click on the item, and the gem is used to charge it. It might that you are not seeing anything in the list because all of your magic items are charged up. Also keep in mind that only offensive items will lose their charges. Pieces of armor or clothing always remain fully charged.