Does the trade good bonus apply to the race that builds it?

The originating race benefits from the trade good, even when traded.  It is lost if the planet which houses the trade good structure is captured.     The converse is true.

Let me see if I got this correct. If I sell/trade a trade good to another race I still keep the benefits and the race I sold/traded it to gets it also?

Yes.  Though there is one caveat and one exception, I believe.

The caveat is that the race that built the trade good must maintain ownership of the planet on which it was built.  Further, a race that conquers or otherwise gains ownership of that planet also gains the trade good.  I must confess, though, that I have never checked to see if a trade good that I acquired by conquest appears in my list of trade goods that I can sell.  I think it does, but I cannot confirm it right this instant.

As for the exception, it is the speed increase trade good (galactic accelerator?).  That is, the builder gets it and can sell/trade it, but I think it is a known bug that the one getting it by purchase or trade was determined NOT to gain the speed increase.  In short, it turns out to be a con!  (though apparently an unintended one)

You got it right LZ.  So does LT.  And...Whoever owns the planet with the trade good structure owns the trade good no matter who built it.

I have never tried to sell/trade them. Can you sell the same Trade Good to mulyiple races?

Yes.

Dang...Time to stop hoarding those things to myself.  I had no idea you kept the bonus when you traded them.  Now I have even more reason to build some resource buildings, and do a little trading.

I'll sell to my allies.

Trading is a good thing.  When I first started I blew it off.  Wonders are a must thing.

Allies?  You don't need any stinkin allies .