I have to set my hotkeys at the start of each game. Is there a way to save them so I do not have to set it every game? Thank you.

Usually they stay saved so it's strange what you are asking.

solution to your problem is to Open your AOE in your PC(not multiplayer).
Click on options and create a "new" user and delete the rest profiles.
and then go on to your hotkeys and set as u like.
U can then try playing some multiplayer games.hope everything will be ok after it:)

Also 1 more thing
You can run AOC on voobly " Single player mode " From options ,set your hotkeys , then go back and take " Single player mode " Off and it will be saved