RO water is generally the best filter method for getting stuff out of the water. In this country we don't usually have to worry about getting sick off the "wrigglys", but chlorine, lead, insecticides and maybe some TCE are a concern, and all of those are still there after boiling. But RO filtration is so good at getting stuff out, it even gets out the good things, like minerals that your body needs. This is also why RO water is "aggressive" and causes reactions that can leave spots or scale on some materials. So read up on it, and maybe a daily multivitamin would help offset the reduction in minerals.
As far as using it in the fridge, check your fridge manual to see if there are any warnings against this. 2 concerns that come to mind are whether the materials in contact with the water are OK with purified water, and making sure the RO water pressure is high enough for the solenoid valve to work properly. If both are OK then I think it's a pretty good decision to pipe it to the fridge rather than have 2 different filters at work.