When they became a little older, the taste doesn't bother them anymore--it just tastes like the sour candies that kids like to eat these days. And now they are even older and can swallow pills, so we don't have the issue anymore.
For the water, yes, sterile water or sterile saline (isotonic or hypertonic) is fine. Our own family uses boiled distilled water. We boil it because 6 hours after you open the container for the first time, PA and other stuff will start growing in it. Of course, you want to cool the water before using it in a neb.
I am afraid I have never heard of demineralized water. I know that distilled water is just H2O with absolutely nothing else in it.