I've always shot with both eye's open, it feels natural for me.
When I'm shooting a scoped rifle, I have to consciously force myself to close my left eye, but I know I've definitely shot plenty of times with both open, but my left eye just naturally cancels it's self out I guess, so it's no real problem for me either way. It's actually seems kind of advantageous though, cause I feel like I'm able to acquire my target a little but quicker when hunting.
As for shooting single handed, the loads I shoot are not that which anyone would want to deliberately shoot one handed. Many moons ago I did try once to fan a DA SBH revolver, and with full house 296 loads. That was a big, big stupid mistake. I ended up getting my left hand ripped wide open. I think either the sight blade got me, or maybe it was the hammer, either way, it sure didn't go well. I thought I had things all worked out, don't we always though when we do something stupid? I'm just lucky I didn't shoot myself in the foot.
GS