How much will a single-core processor hold you back in playing this game?

A single core processor will reduce your frame rate compared to an equivalently clocked dual core processor. However the game is still perfectly playable on single-core processors using lower detail settings.

How does the DX10 version compare to the 360 graphics? And DX9?

The 360 and DX9 graphics are roughly equivalent. The DX10 graphics add a few additional features and image quality enhancements on top of this.

What does the internet activation entail? Why do you have it?

It's an anti-piracy measure that we unfortunately have to implement to curb illegal game distribution as best we can. It's a one time check and then you're done.

Is there another way to activate the game if you don’t have internet at home?

No, you must have an internet connection to install and activate the game. It is part of the system requirements because of this.