Arthritis translates basically as changes in the joints, and over time and over use of the body, certain joints will change in response to the physical stress. The Rheumatoid Arthritis your mother had is genetic, Osteoarthritis (that comes from normal aging and using the body) is not genetic and can be controlled by diet and exercise.

As far as your water softener, the water is "hard" when it is overloaded with heavy minerals, and the purpose of the softener is to take those away, so that the kidney and liver are under less stress. I do not know of any studies per se, but I can tell you that the body will react adversely to either extremes ie too many minerals in the water or too few.  It is likely that the softener is removing too many nutrients, some of which you (or your cat) may actually need to maintain bone and function in the body.

Check the filtering system and call the company and ask them how much they filter out.