Ok so Steam and Origins are like stores that sell games and allow you to access them. You have an account on something like Steam and basically you use that account to play games. Battlefield 3 is not on Steam, it REQUIRES Origin to play. So basically regardless of where you buy BF3 for the PC, you will all play together.
As for the add-ons, I actually only have the main game. I think people either play the original game or the newest add-ons. I have heard some of the earlier add-ons no one plays. I have the first add-on actually, the Karkand one. I didn't see many servers.
I bought the game the first day it came out and it was worth $60 for the original game, I would say the game+dlc is easily worth the $35. Good luck on the new computer! You can purchase it now and install it later.