Doubt it is a keyboard error simply because you can type on that keyboard here.....
Have you tried repairing the game in origin and then totally shutting down the computer and restarting from total power-off condition??
That sounds like a keyboard issue. What it seems like to me is that hold ALT is turning off a modifier to the WASD and TAB key, allowing key presses to be registered normally. So windows/DAI seem to be registering that ALT is always be held down, and pressing ALT fools them into thinking ALT has actually been released.
I would try another keyboard if you have one, to see if that fixes the problem. Or try popping the key off the keyboard and cleaning underneath it, making sure there is nothing under there.
Does it work with both ALT keys? It may just be one that is malfunctioning. You go can also to control panel, choose accessibility, then see if the sticky keys option is turned on. If you have any other input devices (like a gamepad) you may want to disable them or unplug them and see if that helps.
Here are some other threads around the net:
http://www.tomshardware.com/answers/id-1784723/alt-key-stuck.html
http://windowsforum.com/threads/windows-7-behaving-as-if-alt-key-is-stuck.85423/
