but I shall try.

After my character has interacted (E) with anything which can be interacted with whilst running (W), the running key will turn into the walking one (Alt), whereas the function of Alt will turn into the one of W. So if I've pressed E after having approached something running (not sprinting) this madness will occur. To fix this I need to interact with something by running again, which will be rendered by the wrong key of course.

I would be very pleased to know the source of this sorcery so that I can counter it in the future.

Try to delete ControlMap_Custom.txt in your Skyrim folder. (C:\Program Files (x86)\Steam\SteamApps\common\Skyrim)

And remap your keys again. !!!!!

The problem is that my only auto-saved files are all affected by this. So if I load any of them up, pressing W will make my character walk... I did what you suggested and ran toward a smelter to interact with it. Which swapped the keys back to their correct functions. But then I entered a mine by walking into it and the keys messed up back again. I'm out of ideas. At this point I'm not even sure what actually causes this instability.

Update: I deleted the file again and now it seems to be working correctly. Thank you.

You could make that file (ControlMap_Custom.txt) read only if you want. (considering that you don't change your keys again !!!!!)