Pills and condoms are the most used and don't require being cut.

The injection. Lasts 6 months.

I don't think so but even the implant and iud/ius don't require surgery, ten minute procedures and the implant lasts for three years, coils between three and ten years.
The best thing you could do would be to talk to your doctor about the methods available to you.

I got my paraguard fitted 16 days ago and so far it hasn't been too bad. I've read so many horror stories but I guess time will tell how I do with it. Prior to that I was scheduled for an essure procedure but canceled at the last minute because it is fairly new. In an essure procedure, microinserts a threaded up through the vagina and the uterus into each of your falopian tubes. There are no incisions. It is done with local anesthetic. You can check www.Essure.Com for more info.

most forms of birth control involve no surgery!

The pills, paraguard, injections, the iud, the patch, contraceptive film, nuvaring, diaphragms, cervical caps, and condoms of the male and female kind are all non-surgical!