Should I play Empire Earth before AoC?

Is Art of Conquest just a sequel to Empire Earth, or is it an expanded version of the original?

Art of Conquest is an add-on/expansion pack for Empire Earth.

Are all of EE's original missions and storyline covered, if I start a new campaign in AoC?

Starting a new campaign - you can play/re-play the original campaigns but also play the new campaigns added in the expansion.

Would I miss anything if I were to completely disregard and never play Empire Earth - only playing the Art of Conquest game?

If you mean to play only the Art of Conquest missions then yes, you are missing out on probably half the game. If you mean to NOT replay the original campaigns (assuming you played them on the original version of the game) then not really. Improvements have been made to the civilisations which may affect your strategy in some missions. However, generally speaking the original missions are exactly the same in the expansion.

I don't know which game user101016 played but it's clearly not Empire Earth / it's Expansion.

Let me get started with a clarification Prior to explaining/answering your questions (sorry for some words being capital, somehow autocorrect doesn't like english on my PC)

Art of Conquest, short AoC, is an Expansion not a DLC. This is a huge difference in my opinion and I'll try to differentiate them a bit more. DLC is nowdays the more common term for any Content you can download additionally (as the Name already says..) but it mostly refers to additions, Content that is directly added into the existing game and expands your experience. For example in an open world game a DLC would add a new Country or Island to explore. In a RTS game like Age of Empires a DLC would add a single new campaign or a few Units.

However back in 2000 up to until roughly 2010, Expansions were way more common. An Expansion was launched roughly a year after the Initial release of a game and was often actually a stand-alone game that required only the main game for a few assets (to Keep the size of "new" game smaller for example). Expansions normally resulted in a completly seperate game you installed. A seperate launcher. A seperate instance of the game with all the new Content directly integrated often with new campaigns or missions. Rarely you were able to Play the "original" Content since expansions often completly overhauled the Balance by adding a lot new Units. For example the Expansion "Forces of Corruption" for Empire at War adds a complete new faction, you have 3 instead 2 and completly new mechanics. It often feels like a completly different game.

I hope this clarified the differences a bit.

Art of Conquest is an Expansion to Empire Earth. It's a seperate game and for example in the multiplayer, you can't Play with People only having normal empire earth. EE has 14 Ages, AoC adds a 15th to it, it adds also space battles to the game, which requires completly new maps. It balances other Content out and it also adds completly new Content for civilizations like priest Towers. Instead of boosting your unit stats with civ Points you can use those to unlock buildings or Special Units. The priest Tower is a defensive building that automatically converts enemy Units in it's range.

Due to this Balance changes it's not (nativly) possible to replay the original campaigns, since you would have Access to this new Content (since it was added later, it of course can't be blocked in a Scenario. Imagine Hasbro Released Monopoly 2.0 with new Content. If you have custom rules for your Family evening, those new conent isn't covered obviously)

However, both EE and AoC Feature completly seperate stand-alone campaigns. EE = Greece (Stone Age -> copper Age -> early medieval). England (medieval Age up to the times of Napoleon) Germany (WW1 -> WW2) and Russia (set in the future). Every of those campaigns cover often over 100 years and most missions are only roughly related or follow multiple persons over time.

AoC adds more "straigh Forward" campaigns that follow only a short time window or an ongoing Story. For example the USA in the pacific in WW2. There is an asian campaign set in the future with Focus of travel to mars and a Revolution there then. And a Roman campaign with the Focus on Ceasar and how he got to power. (AoC has only 3 campaigns)

You are not activly missing anything besides the stories if you Play directly AoC though. I recommend however to Play all campaigns since they are pretty good and often challenging. Please be Aware that cheat codes do NOT work in the campaigns of EE (though there are Workarounds for that. In AoC cheats work)