If you want direct weather information you can call the Belize Meteorology Department and get a recorded weather forecast every day. It has coastal, mainland and mountain forecasts along with sunrise, sunset, tides, sea conditions, etc. Phone from outside Belize by dialing:
011-501-225-2480.
As for water temps - I do not know of any service that tells you this one, nor visibities - perhaps you can contact a dive shop to see if they post such info.