UV light is harmful you your eyes. UV light is the same stuff that gives you sunburn, so you can imagine the damage it does to your eyes. In fact, a common example of how UV light can hurt your eyes is snow blindness. Basically, the UV light cooks your cornea. The same phenomena also occurs to welder's who don't use proper eye protection. You can read about it here
http://en.wikipedia.org/wiki/Snow_blindness
ALWAYS WEAR SUNGLASSES WHEN GOING OUTSIDE. Even when it is cloudy, and not sunny. UV light still penetrates the clouds, which is why you can still get sunburned even on a cloudy day.
After I had my PRK Surgery (very similar to Lasik) I shoot all the time with sunglasses on, and it hasn't hampered me any. Being that I wore glasses before the surgery, I was used to shooting with eye glasses on. I will say that using polarized sun glasses makes it hard to look at the LCD, as you can see imperfections in the LCD screen very easy, and the image in the LCD doesn't stand out so good.
I occasionally shoot with 2 eyes open. I do a fair amount of shotgun sports shooting (skeet, trap, 5 stand, sporting clays). When shooting a shotgun, you are taught to shoot with both eyes open, looking down the barrel, so it comes natural to me.