Need to restore, but have a question

I sold my computer last week in preparation for the MBP that I want to buy. I saved my backup info for my iPhone. I haven't uploaded that backup info to my wife's laptop (which is tied to her iPhone), but I just hooked up my iPhone to her computer and it let me back it up.

I'm wanting to restore the phone to the backup I just made. When I hit restore, it says it will restore to factory settings... but I want to restore with the backup. After the restore to factor, will it let me restore from the backup? How does this work?

The reason I'm doing this is because this is the third time that the right side of my screen has gone dead on me. It's not touch sensitive. I took the phone in last week because of it, and they said it might be a software issue as I couldn't remember if it affected the entire screen or just the SMS application as that is where the issue happened and then I restored without checking the rest of the phone. Well this time it is confirmed, the entire half of the screen is dead until I restore and then it's fine. I'm going to take the phone back in today to hopefully get them to replace my phone this time. Hopefully they will because they just announced that they are getting rid of their stock of the 32g 3GS which is what I have. Any tips on convincing them to replace it for me?

 

Ok, so I replaced my iPhone 3GS after backing it up. I got home and restored from backup, yet all of my apps are gone. Everything else like my notes and pictures are here, but none of my apps are. I have 3 pages of games... some of which I have spent months on and the saved games are very important to me like Plants Vs. Zombies and Hookchamp.

How do I get my saved games back? Do I need to sign into my itunes account on my wife's computer and sync the games to my iphone and then restore again and my saved games will be back?