That's a reasonable way to go. Most personal injury cases (90%+ in most states) are settled either before filing the lawsuit or after filing the lawsuit but before the trial. If you can get what you want without going to trial that's a good result for you.
Yep. You are entitled to compensation for all the harm suffered from the accident — your medical bills, of course, but also compensation for lost wages/income, pain and suffering, any property damage (which would include even clothes damaged in the accident), etc. Since the driver would owe you for all that in a lawsuit, the insurance company pays for that.