So is a Khajit a man who was bitten by a cat then?
And is an Argonian a man that was bitten by a lizard?
I am totally confused and this is ruining Skyrim for me now.

They're just humanoids: different races that share a number of traits similar to a human. Think of Wookies or Tuskan raiders from Star Wars. It's the same overall concept.
No, Khajiit and Argonian are hackneyed, lazy attempts to create a new, 'interesting' species by adding animal elements.
I don't see how adding animal characteristics
in tandem with other unique qualities about their race and cultures makes them a 'hackneyed, lazy attempt' at making the different playable races more distinct. They've been fleshed out very well in the lore over the last few games.
So would you become a ManManWolf or a WereWereWolf?
This is all too confusing.
What if a Werewolf bit a Goat? What the heck would that be? A WereManWolfGoat or something?
When a werewolf (a man/mer/humanoid who turns into a savage, half-wolf, half whatever it was before) bites something else, it transmits the disease that causes them to transform into a werewolf. It doesn't matter what the other half of the werewolf was (orc, khajiit, high elf, imperial; it's all the same) and it doesn't matter what it is they're biting. It transmits the same disease that does the same thing regardless of what species you are.
Some species may be resistant or immune to the effects, though (due to perhaps lore, gameplay constraints, or both) such as primitive animals and such, but just to use your hypothetical situation: if a werewolf bit a goat and that goat DID become infected, it would turn into a half goat, half wolf creature. If that creature were to then bite an argonian, and the argonian were to become infected, he would transform into a werewolf.