EIA-914 (Monthly Gross Production), while not nearly as timely as the weekly storage report, can re-set supply expectations if a particular field is pumping a lot more or less than the market thought.
Depending on trading style, you might also follow daily or weekly CDDs, HDDs (cooling and heating degree days), FERC's nuclear plant outage report, and power burn and pipeline flow estimates.
Sometime in 2015 LNG exports from the Gulf of Mexico will also be a closely watched number.
As a general comment about weather, unless you enjoy following it closely and have the bucks to spend on private services, you can probably get all you need from the NOAA/NWS websites. The big boys are buying reports from multiple services and some of them even have a meteorologist on staff to interpret and analyze the reports. FWIW, I think the 3-5, and 6-10 day forecasts are updated every six hours. Not sure about the 11-14 day and 15-30 day forecasts.
As a longer term trader, price and the weekly storage report incorporate most of what I need to know about the weather. I'll pay closer attention if we're in a period of abnormal and persistent heat or cold or if a hurricane or sharknado are about to hit, but those are infrequent events.