Yeah, it is a bit frightening when I re-read what I've written. I guess it was clear that what I meant is that you should never use sea water to prepare limewater. My guess is that we do, but that if we have sufficient mixing the microscopic bits of precipitate go right back into solution. That condition is never satisfied in your tank outside of the tiny area where a droplet of limewater hits the tank water. It is of course satisfied very well in limewater solution.
Magnesium hydroxide is rock insoluble at limewater pH values.
There is a call for such things occasionally, but I'm personally unaware of extremely well debugged plans laying around on the web somewhere. Regarding the questions asked, my personal preference would be to use an overhead mixer rather than a stirbar, and gravity output from the lime container. Obviously some sort of metered input. I think that he used a modified fish feeder to slowly meter lime into the mixing container. There are some advantages, though.