Re: Does eye dominance matter?
With a handgun it shouldn't matter as most people will naturally align the sights to their dominant eye when shooting with either hand. However with a long gun (or bow) that's obviously not possible since the angle of the gun is not flexible as it's connected to either the left/right shoulder.
Some people do learn to shoot very well swapping sides, but it typically takes awhile. In fact over time many shooters that make the switch shoot better than they ever did the other way. The quick fix is as George mentioned and using tape/blinder/etc. making it so the dominant eye cannot see the sights forcing the non-dominant eye to take over. With open sights there are other options, a longer front sight hood can obscure the non shouldered side eye from seeing the sight, or with a shotgun they also make a little "blinder" that goes on the vent rib by the barrel to do the same task. That's typically less "obstructive" to your peripheral view than the tape on the glasses but still forces the brain to accept sight input from the non-dominant eye and is probably a better option when having to spot and shoot moving targets such as trap/skeet/etc.
A lot of people can get away with it on scopes as well, without switching their brain over time will learn to accept input from the non-dominant eye. It's a crap shoot though with optics I know people that have no problems shooting shotguns/handguns both eyes open but their brain struggles shooting a scope both eyes open, some people seem wired for the brain to want the magnified input as dominant and some the non-magnified regardless of their eye dominance itself. Some of the most frustrating shooting with a scope is if both your eyes are very close to being equally dominant, your brain will flop back and forth focusing between the magnified and non-magnified input. Sometimes a person over time again can train their brain to stay with the magnified input, others just end up using a blinder or glasses tape.
The scotch tape is the best option to start with though, it's cheap (if not free) is easily adjusted and fully reversible without consequence.