Cuz Ur Stupid
Technically yes you can. Your PSN account is a Sony-wide service regardless of the device you are using it on. You can redeem a PS3 code while on the PSVita's own PSN 'app', and it will be added to your PSN account. However of course, that will not let you use the PS3 content on the PSVita.
PSN Codes: These are like serial numbers, lets say of you went to the shop and you bought a map pack for a game such as call of duty then you would recieve whats called a PSN Code, this stands for playstation network code, this is basically a key that when entered in the online store it will let you download the product that you have bought.
It's actually known as a PSN ID. http://www.us.playstation.com/PSN/mediago
You can't. PSN stands for PlayStation Network. You can't connect a DS to PSN.
WHAT DOES PSN STAND FOR OR MEAN
by PSN you need a account to PSN
The PSP-3000 can access the PSN, yes.
No, PSN is free and there is no indication that is going to change.
Yes, there is a PSN link on the Vita's LiveArea.
The password is whatever you set it as when you registered your PSN account.
System activation is where you log onto your PSN account on a PC, and register your devices (PS3, PSP, PSVita). Only systems you have activated in this way can log into your PSN account and use your PSN data.