The NES (original, not the top-loader) had a design flaw with the pin settings. Since they're springs, eventually it will make it more difficult to start up a game. "Blowing" in the cartridge won't do anything if the connectors in the back of the system won't work. It's why when they released the top-loader version they said that it wouldn't have the "blinking screen" 99% of the time, because you weren't bending the connecting pins. What do you think you're doing when you push the cartridge down?
As for your 360 issue, I believe Microsoft's warranty is 1 year from manufacturing, so just give them a call about it. Replace the harddrive by putting it into your 4GB model, and use the 4GB model with your harddrive until your original 250GB model is returned fixed from Microsoft.