After i research it, the wormhole will become visible on the map?
No. If there is a wormhole on the map, you have to find at least one of the ends, and send your ships through it.
if there is only 1 star in the system, the wormhole will not be available?
Also depends on the map. Some wormholes are on opposite sides of the same system, others go to other stars. Some maps have none.
Will the trader go through wormhole?
Once you've already sent ships through it, yes.