I just downloaded the 1.13 patch and whenever I try to connect to any server it tells me that it can't identify the game version. Any reason for this?

Run the game in Administrative Mode.

That seemed to work. Now whenever I'm in the game it seems to crash shortly after. Not just the game, but my entire computer freezes and crashes.

Alright... Try running the game using different modes. Follow the steps below:
1) Create a shortcut to the game on your desktop.
2) Right click on the shortcut and choose "Properties".
3) In the target box after the last quotation, press the space bar once and type in the command lines below:

Windowed Mode:
Your target line should look something like "C:\Program Files\Diablo II\Diablo II.exe" -w