No, your insurance rates shouldn't go up due to this accident. (Important words: Due to this accident.) While the rules and laws do differ by jurisdiction, this rule stays pretty much the same everywhere. If it wasn't you fault, it won't affect your rates on a personal policy. (No promises on a commercial policy, they're not rated the same way)
And no, using your Collision coverage (or Uninsured Motorist, if it covers your vehicle) does not make the accident your fault. That's one of those insurance myths that just refuses to die, just like the red paint surcharge. Collision does cover at-fault damage, but it also covers damage by hit & run, impact by bicycle / shopping cart / roller-blader / pedestrian, and it covers the vehicle if it spontaneously flips over onto its roof for no known reason. (I've never heard of that last thing actually happening, but Collision will cover it if it does.)
You can take the other idiot to court if you want, but you're better off claiming through your policy and letting the insurance company go after her in court. Why? Because they do this stuff for a living every day, and you're thinking of doing it for the first time. They have an entire team of people (called the subrogation department) who do nothing but take people to court for this stuff all the time, and you have... what? Exactly. You pay your insurance company, you probably pay them a whole lot, so let them earn their money for once.