For the question of baking soda in beans, i know it will make them cook faster, it also works to make fresh vegetables like string beans and artichokes stay green, but what i also know is it's not good from the nutritional point of view - from what i always heard, it makes the vegetable lose more nutrients into the water.
I'm not sure why you would want to use mineral water, shel. I would think mineral water would have more minerals so it would leave more limescale on the pots. Is the water where you live so bad?
I stayed overnight in a place outside Rome once, where they bottle the water that comes from the springs there, and it was bizarre - the faucets were all full of limescale, the chrome peeled off, and the water came out of the faucets with bubbles in it (no co2 added!) (Most of the good mineral water here, ferarelle, st agata, nepi, have no added CO2, because it comes out already bubbly.)
I can't imagine it would be better for cooking. In fact, some people have demineralizers in their homes.
and for a funny video on water take a look at this http://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CDYQtwIwAA&url=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DNqXFrs6quvE&ei=JgN6UvD1D8TesgbdkIGoBA&usg=AFQjCNFBObYc13cVfIMHrJmPGFmIcINW5Q&sig2=jrMu8nW7L5KzVwIihP9vuA&bvm=bv.55980276,d.Yms