Oren. I made myself ABUNDANTLY clear that she wasn't mentally unstable because she played with Barbie dolls or dressed in black.
For the third and final time, in the Anime she is portrayed as having a taste for dolls which are not normal (with broken limbs and eye patches etc) and wearing black yet intriguing clothing, not because she wants to be with the IN Goth crowd, but BECAUSE of the trauma which was caused her over her parents' murder. This is shown when she is being introduced in the Anime for the first time.
Also Nanami, she does know he's false. She is just in denial about it because of her "mental instability" over the loss of her parents.
Why does she know that Kira is false? Simply because she is the 2nd Kira, plus she has the death god eyes, which make her more powerful than Light.
She can easily put two and two together and figure out that Light is a false god, or at least inferior to her but she doesn't. She loves him because he killed the killer of her parents and is infatuated with the idea of him before she ever even meets him.
Sorry if I sounded like I was making you feel stupid before. Bad day lol.