Did Rheya have a mental illness?

Real Rheya did. She was suicidal.

Every man has two nations, one of them is France. (B. Franklin)

Rheya was created by Solaris from the subconscious of Chris. She was exactly how Chris's subconscious projected her at the moment of her creation (which is why the two created Rheyas were so different), she was defined not only by memory "playback" but defined by his emotions of guilt, sorrow, anger, etc., as well. Interestingly, Rheya seems to have a sense of this "incompleteness" of her being and is driven mad by it.

She has all the neuroses that Chris believed her to have.

Yes. The real Rheya's inability to hold a job, mood swings, reluctance to have a child, and eventual suicide are classic signs of bipolar disorder.

Are you not entertained?!