I had 200 Atari tokens, so I unlocked Caitians, can I now re-create one in the generic Alien section so I can assign the 4 starting traits as I wish, or no?

No. Purchased races cannot be used in the Alien creator.

I still say that when you purchase the race it's parts should be unlocked for your own use in making other aliens. This is really a major issue with Catians, though, as basically every other C-store race is easily recreatable with out buying them.

I did this with a Klingon so that I could use all the tribbles that are available in game.

personally i'd settle for having a caitain with a head of hair.

<-This!

Acutally this is the major reason for me not buying those races. The traits are pretty uninteresting for me. I want only the bodyparts, because I am a "avatar changeling". I can't play the same toon for too long, so I have to change it. The tailor idea and the totally changeable alien race for STO had me already for the first moment. STO is actually the only MMO, where I have my oldest active toon playing.

I was amazed while learning, that a lot of other players think the same. So, if Cryptic would unlock the bodyparts for the featured races in the C-Store, there would be a lot more players willing to buy those. It would be a win/win scenario. We'll get more bodyparts, and Cryptic would get more money.

I think that it currently is not easily possible might also have to do with their costume system that sometimes offers unique changes depending on your racial choices. (You need to handle stuff like tails, for example). But that's just a guess.