I am right-handed, left eye dominant. There are actually more folks with this paradox than you might expect. I have shot right-handed long guns my entire life (bows also). I've only owned two left-handed guns: 1) an early 70's vintage Remington 1100, and 2) a Winchester Model 70 in 300 Win Mag. I still have the Remington, but sold the Winchester several years ago. Honestly, after having shot right-handed guns from my left shoulder my whole life, it felt odd cycling the bolt with my left hand (go figure). Bottom line: Unless I found a can't-pass-it-up deal on a left-handed rifle/shotgun, I wouldn't think twice about accepting a right-hand.
I shoot a pistol right-handed, but cock my head over to the right so as to place my left eye in line with the sights. I know, weird. But, the moral to the story is we all find a way to accommodate the tools we have. For the record, I can't recall ever having an incident whereby the right-hand ejection of a spent casing/shell caused me any distraction whatsoever - and that includes thousands of rounds through semi-auto shotguns and rifles. And, regarding cycling a bolt action with my right hand? It's just not an issue. I remove my hand from the forearm supporting position, maintain approximate sight position by holding the stock firmly on the pistol grip with my left hand, cycle the action and then return to the forestock support position. It's really not a thing (at least for me). I suppose there might be some rifles that are so front-heavy that it might prove difficult to do this, but I've never run across one yet.
Steve