it ok to let the roots grow in water thats the method called (DCW), just make sure the water isn't too hot or it could kill your roots.
a good water temperature is between 65-70 degrees,
dont let the roots grow into the same container as your water pumps because,
-it will be engulfed with root mater and suffocate the water pump -seriously...
-and the pump adds heat to the water.
the bigger the air stones the better. i found some sweet circular ones that are 5" at a pets store. PERFECT for the bottom of 5 gal buckets.
something to remember, if your air pumps are in a hot location they will be pumping hot air into you water, causing it to heat up.
check out my DWC grow journal (link is in signature)