Regret to advise wood under a battery is an old wive's tale. If you are charging it to the extent heat is generated, something is wrong. In any event battery cases are good insulators, heat rises and lead is the most conductive part of the battery so heat will conduct up through the terminals.
Heat degrades batteries and so does overcharging. Cold temperatures are good for storing batteries - not so good for performance though. If you look at the stats you'll find batteries need replacing more frequently in hot climates because of progressive separator degradation, which also happens when you overcharge them to the point of 'gassing'. Equally, never or seldom charging leads to the dreaded sulphation of the plates - irreversible loss of capacity - so it's a compromise.
As for rainwater - it's better than tap water but deionised is preferable and distilled is the ultimate. Unless you are in really hot weather or overcharging you shouldn't be losing water as the sulphuric acid electrolyte has an affinity for water and should reach and maintain equilibrium concentration.
