But the vendor agrees to let Amazon set their prices in return for, essentially, being an Amazon-sponsored product. The vendor doesn't have to participate, in which case it can still set its own prices.

I can see the argument that it's price fixing if Amazon & the 3rd party both agree to sell at the same price, but even then I'm not sure it's anti-competitive or anti-trust when some other 3rd party can come in, see the higher prices, and decide to undercut them. Not unless Amazon get's all sellers of a product to agree, but the article doesn't directly allege that.

You sold that batch of mugs, but that's it.

Nope. You still have the right to price your own future batches of mugs.