How does manifesting work?
Yes, you may look at the card, since you control it: 



  707.5. At any time, you may look at a face-down spell you control on the stack or a face-down permanent you control (even if it’s phased out). You can’t look at face-down cards in any other zone or face-down spells or permanents controlled by another player.



No, your opponent may not look at the card, since 707.5 only allows the controller to look at it. 
N/A see #1
No, a manifested card does not go on the stack; manifesting is an action taken as part of a spell/ability resolving.
See #4


Also, see rule 701.32 for more information on Manifest.