How can I prevent unpleasant smells and organisms in our water tank?
Besides the suggestions above some things you might try:

Oxygenation: a small pump set up in the tank pumping the water into the air gap above water float level you have or an air pump pumping air into the water might help with some of the issues. The pump can be placed on a timer to cut down the electrical expense.

Refreshing the water on a regular basis.  A tank I set up for a Solar off-the-grid type situation pumped continuously into the tank during the day, (but with solar powered pump, so pumping was free), and the tank had an overflow outlet near the top which could be tapped into drip irrigation system.  Water was exchanged continuously, keeping growth down.  An alternative to this would be flushing and refilling the tank on a regular maintenance schedule. 

Lastly, and possibly the best solution, would be simply placing a filter canister with a taste/odor filter cartridge between your tank and rest of the home.