Non-DirectPlay game path is 'e:\steam\SteamApps\common\Age2HD'
Verifying 'ebueulax.dll' exists...
ebueulax.dll was not found in its proper place.

Your version of the game is not compatible with Voobly.

You need to reinstall the game in its default location. Please follow the steps mentioned here to perform a clean installation of the game:
http://support.voobly.com/display/vkb/KB1054+Clean+installation+of+Aoe2HD+with+compatibility+patch

Follow the steps as is. Do not skip/jump steps.

e:\steam\SteamApps\common is where my games go to though. Is that ok, or is that the reason it didn't work the first time?
Can I somehow save my hotkeys in AoE2HD?

Nope, reinstall the game to its default location as outlined in the article.

You can backup your .hki files from your current HD installations (copy them to a safe location elsewhere) which you can restore later on your new install.