Does it seem fair that even though Ezri Dax is half "Dax" she is still a Lieutenant (J.G.)? Why doesn't she retain her previous host's rank (Lieutenant Commander) and position. After all, she has all her knowledge and experience. Does the symbiont have any rights of its own?

The rank, experience, and accomplishments, of the present hosts, are what presents the joined individual at the moment. It seems to me, that all that Ezri inherits other than the merging with another personality, is just the memories and knowledge.
I seem to remember from a couple of the episodes about Dax, is that the current hosts does not get accounted for the behavior of the past host. Something like that.
The LT. j.g. is what Ezri earned. The Lt. commander rank died with Jadzia.

Tough but fair. After all, a long-lived symbiot could, theoretically, enable a right prat of a host to be Fleet Admiral or something.