How do you interpret "Part of the action you are attempting to perform stealthily", then? To me, it seems to indicate that the type of action you perform modifies the form of stealth that is granted. However, it's ambiguous enough that I won't cling to that interpretation.
To be more clear, Stealth checks are made when you're trying not to be seen or heard while taking actions.
If you're standing there talking to a guy -- or just standing next to him -- and want to palm a key off the table without being noticed or drop a pebble without him seeing or hearing it... that's Sleight of Hand, a thievery check.
If you're hiding in the shadows and you want to put down a heavy object or whisper to another character without revealing your presence, you make a stealth check.
Stealth checks relate only to gaining and maintaining the condition of
you being hidden, not to conceal or misdirect an audience from noticing your actual actions. Stealth is all or nothing; the creatures are aware of you or not aware of you.
The power says nothing about getting around the normal rules for stealth. All it does is eliminate the movement penalty for your check.
Correct. If you want to hide while moving between cover, such as to cross a lighted hallway while maintaining stealth, you need the level 6 rogue utility power "Chameleon". Shadow Stride is even MORE powerful; it essentially gives you the benefits of both Chameleon and Fleeting Ghost simultaneously, so you can retrain one of them if you already had it.
Edit: Oh, actually, hang on. Chameleon is an interrupt. You can't use interrupts on your own turn. So Chameleon only allows you to retain stealth when something happens during another creature's action, like a creature moving so you no longer have cover against him, or somebody turning on a light to get rid of the shadow you were hiding in. Shadow Stride doesn't give you that functionality, but allows you to retain hiding while intentionally crossing a lit or uncovered area. Eh, whatever.
I never claimed you could. With a normal move with Stealth check, any time you break out of cover/concealment you are visible. If you re-enter cover/concealment by the end of your move, you can make your Stealth check to become hidden at that point.
Correct, but at that point you've shown your presence. You can hide again, but now the enemies are alert.
If you don't have Shadow Stride, your best bet is to try to get that Distracted clause in there -- which is totally up to the DM -- by doing something to make everyone turn around, like throwing a rock or casting ghost sound.