When limewater is exposed to carbon dioxide it will begin to form calcium carbonate. This is probably what you are finding at the bottom of the container. To reduce this somewhat be sure to mix with cold water, keep the container covered as much as you can, and don't overstir when mixing it, never shake it to stir. It cannot be prevented completely. Adding a few teaspoons of vinegar to the mix will also help.
I would recommend cleaning any precipitation out of the jug every time you refill it. Never allow it to enter the tank.
The amount and frequency of dosing limewater completely depends on how well it is maintaining your Alkalinity and to a lesser degree, how well it is maintaining the Calcium level. If you have a lot of corals and they are lowering your Alkalinity quickly then you will want to dose limewater 24/7 every day to replace 100% of the evaporation. If you have only a few corals you may only need to dose ove a week. In my personal opinion I believe it's best to dose it every day either 24 hours or the 12 hours opposite the lighting period and control the Alkalinity level using the amount of lime added to the water.