After the recent hate crime in the Village (just an example) which I thought was rather safe I wanted to do some research into which intersections/streets/'hoods have higher crime rates. I seek violent crime statistics (such as shootings, stabbings...) for 2013 and am particularly interested in the following neighborhoods: West Village, Willamsburg, Upper West Side, Gramercy, Chelsea and Lower Manhanttan. Also, data covering crimes against women and hate crime would be welcome.
The City of New York publishes it's crime stats by Precinct on a weekly basis.
Even the most cursory of Google searches meanwhile, will reveal dozens of visualizations or other attempts to parse and crunch this data.
That said, all of the neighborhoods you listed are extremely safe, both in general, and in terms of hate crimes. The reason that the recently reported incidents which prompted this question were so incredibly shocking is because they were so shocking. They represent the sum total of serious incidents of that nature in those neighborhoods in recent memory for the most part.
New York City is still probably the safest large city in the USA. While more recent FBI data has shown that rank falling a bit, most of that is attributable not to a rise in murders, rapes, or assaults, but to a slower than national average decline in robberies - most notably, events like cell phone thefts on the Subway. While this is certainly reason to be alert and aware of your surroundings, I'd hardly call it a reason to be vigilant about any neighborhood in Manhattan, or, for that matter, to be especially concerned about hate crimes.
One thing that you will notice is that Greenwich Village and Midtown do tend to rank poorly in the cities crime statistics. This is for two reasons; one, the aforementioned high number of petty robberies is at it's highest in areas with lots of tourists and a large commuter density - especially Midtown, which has a relatively small resident population, which drives the per-capita numbers sky high. The other issue, is that both neighborhoods represent a significant portion of the cities nightlife, and the various crimes that come along with that - which is to say, assaults from bar fights, rape (though usually not of the stranger in a darkened alley variety), and yes, robbery.
As a rule of thumb, as a tourist, you'll have nearly no reason to go anywhere where you should be legitimately afraid for your safety. Neighborhoods like East New York, Brownsville, and Hunts Point have nothing to offer you. If you stay out of them, and don't go looking for trouble, you'll have little to worry about in New York.