How does one upgrade the workshop to Level II or Level III? Is a special item needed to do this?

I'm not sure if it's possible on the first play...
I suspect you'll have to unlock more advanced corporations first.

There are also projects that you can find scattered on planets, or obtainable from sellers... maybe the workshop upgrades could be found in those ways too...

So the projects on other planets could be the key to unlock the "Advanced Workshop" option? Right now it's not available in the Workshop.

I started another campaign, after finishing the game once, and I just bought a "Workshop II" project from a merchant.
Also I got a "Clone lab III" found in a wrecked ship.

I didn't see those options in my first play, I'm not sure if was because I changed faction, anyway, those are not "upgrades", are just new modules, so you can build all the versions at the same time, or just destroy the old ones. (The more advanced can do both the old and the new stuffs, like the advanced research.)

Maybe I was wrong about what you can find in planet missions, for now I found these things:

- upgrades that stays there even in future new campaigns (more storage, faster mining missions while you are not onboard, etc...)
- upgrades for your avatar (damage bonuses for weapons, protection bonuses, points of interest on missions, etc...)
- log recordings for the background story
- new projects for weapons and armors
- info about the area (where to find types of aliens, where to find components, etc...)

AFAIK, you just build it (if you got the right resources). You don't upgrade. Build one, and scrap the old one.

To get upgraded rooms you have to either find them or buy them. To find them invade wrecks and pirate ships using the Ship Assault. To buy them look for Merchants. Both ways will be random regarding what you get.