When trying to enter quotation marks into the console (which is required for some console commands) nothing happens. It's like the key isn't recognized by the game at all.

could be a location thing, which key are you using? In England " is Shift+2 where I believe in the US it's Shift+/ try using the alternate key combo (for me it's Shift+2 in the console, as it should be)

EDIT: actually, on second thought, I believe in the US " is above the apostrophe... not forwardslash :/

Uhm...

Yep, Us layout for keyboard should have " above apex ' and near the enter key... but i never heard of this issue before.
Have you remapped your keys in game?