Yeah, let's talk about Molly.

What we know:

She is a pathologist: she spent most of her twenties in school becoming a physician, likes investigating medical mysteries (cause of death), is fascinated by human physiology in corpses, works around the dead mostly, and is an extensive resource for Sherlock Holmes.

She likes sociopathic men (particularly if they whip corpses, apparently).
She likes mismatched patterns and bright girlish but practical fashion.
She is socially awkward, and has a morbid sense of humor, but is very emotionally intelligent ("I know what that means, looking sad when no one can see you").
She has rather low self-esteem ("I don't count").
Her father is dead.
She doesn't have a problem manipulating the law for friends (stealing a body for Sherlock).
She is friends with John, Mary, Sherlock, Mrs. Hudson, and Lestrade.
She is very loyal.
She is seemingly anxious to have a man in her life, so she's probably quite lonely.

What else?
 

Liberty wrote:

How do we know she was in school (medical school?) for most of her twenties?   Most courses are five years, so she'd be qualified at age 23 if she went straight from high school.  Do we have some other background on her that I've missed?  (Or do you mean her specialist training?)

I think what I'd change about your list is that she is more Sherlock focused.  She doesn't steal bodies for any friends, I don't think, but does it because it's Sherlock.   She's not just looking for a man in her life, but secretly wants it to be Sherlock, so looks for a Sherlock substitute and is disappointed (although I think she's coming to terms).  Although she appears very sweet, we have seen her be violent a couple of times, which was unexpected. 

This isn't a spoiler thread, so I won't say much, but it will be interesting to see where they go with the character in S4.  I don't think we've had the full story yet. 
 

Oh, haha
I totally didn't look up how med school is in Britain. Specialist training is separate from a complete medical degree for the Brits?

You're right. Sherlock is definitely her exception-to-many-many-rules.

The violence is kind of fun when it's there because it's generally justified. Though, slapping a drugged-up man much bigger than her was a bit rash/brave.

I like how she's progressed to the point where she is still a good friend to Sherlock and still very generous and kind, but no longer takes his bullshit.

No, we have definitely not had the full story yet. Gatiss and Moffat need to do justice to their first original character.
And by justice, I don't mean get her a solid romance (I mean, possibly). They need to give her a complete arc, no matter how small.

How would you like to see it finished?