When storing premixed water, do I need to have an air stone bubbling this water all the time?

Not really. I do keep my stored water with both a powerhead and a heater. Make sure you keep it in a dark place out of sun light. How much are you storing at a time? Make sure that its close to the same prams as you tank temp, sg, ect. before you do your water changes. :raining:

No, but it should be as airtight as possible. I would only circulate the water before you use it to ensure its at a stable PH.