I'm a southpaw and there are definitely some things in kitchens that are designed for R-handers more than L-handers - serrated knives have the serrations on the wrong side for things like bread-slicing, manual can-openers are a super-pain, ladles pour the wrong way, measuring marks on cups or pots often can't be read while holding in the left hand, and spatulas are always slanted the wrong way (is this the problem with the batter-folding that you mentioned, by any chance?).
I bet i organize my home kitchen differently than a R-hander would. I prefer my spoons and ladles and tongs handy to my left hand while cooking. I stand to the right when using the controls on a right-handed appliance like a microwave or a toaster oven, while a R-hander stands comfortably in front. If I'm chopping a bunch of veggies, I want to direct the end results to the left in most cases, not the right.
If you have a tight kitchen and it's set up for right-handed work, a southpaw might find it more awkward to do the same tasks. Their ideal work flow will be a bit different.
There are some nice kitchen tools for south-paws. If I worked in a kitchen I would probably bring some of my own knives with me. Check out web sites like Lefty's in San Francisco or Anything Left Handed in the UK to see why these might be beneficial.
Try working in your kitchen with your left hand for a few minutes at several different tasks, I bet you'll see how some things differ. I don't think you'll find many left-handed appliances, by the way. And I'm not sure why your L-handers seem to be bad at chopping or cut themselves a lot, but you could encourage them to try bevelled blades for L-handers? (thought I wouldn't think this would be necessary, in most cases)