I have a close friend who is highly intelligent and also mentally ill and as such I've done a fair bit of reading on the subject.
Recent studies have shown that schizophrenia is definitely not correlated with high IQ: http://www.medicalnewstoday.com/articles/287555.php
And while depression seems more common in people with IQs over 130, it's also more common in people with IQs under 90. Not to mention which, people with IQs over 130 may simply be more attuned to their "mental weather", as it were, and more capable of seeking out help and describing what their problem is. How do you control for something like that?
Do more gifted people have depression or are gifted people simply better at recognizing the signs and getting treatment for it?
My friend, who has suffered clinical depression all her life (she was on meds as early as the age of two), says her high intelligence has been an asset to her, not a liability. Because she's smart and creative, she can find ways to cope. Frustratingly for her, her mental illness often gets in the way of her native intelligence. She's well aware of how much more creative and productive she could be, if only she wasn't mentally ill.
All of which is to say, just because your child is gifted, that doesn't mean she's doomed to suffer mental illness. I'd be less worried about her IQ, and more concerned about the fact that severe mental issues evidently run in your family.
Also, reading to your child and encouraging her interests won't make her gifted. She is what she is. And providing an enriching environment is simply good parenting.
It's natural to worry about the things that set her apart from her peers. Your instincts as a parent are telling you that this is dangerous for your child. And you're right - any trait that draws attention to us, positive or negative, has the potential to attract predators. But, in this case, your instincts are leading you astray. You don't have to worry about lions in the grass. All you need to concentrate on right now is continuing to nurture your cheerful little toddler. Teach her to read, if that's what she wants. Or don't. Have fun with her. Accept the compliments as they come, but don't take them too seriously.
When the time comes to enroll her in school, if she seems significantly ahead academically, consider getting proper psycho-educational testing done. She may qualify for some enriched programs, which give her the opportunity to hang with similarly academically talented kids.
The most common challenge for gifted kids isn't mental illness, it's finding other kids who share their interests.
By the way, you were almost certainly gifted. Most "super-early readers" are, except when they're dually exceptional and on the autistic spectrum. But it seems that giftedness is less often recognized in girls, especially when they have brothers. Society trains you, your entire life, to step back and downplay your talents and give credit to others.