Re: BatteryWith most battery tools these days I just leave the battery on the charger all the time, as most chargers will just cut out when the battery is full and will cut in again if the battery loses charge. I was told at the tool shop that this keeps them in the best condition. Probably worth checking on the individual machine instructions though...
Re: BatteryIt will depend on the type of battery - lithium-ion batteries tend to hold their charge and if you do allow them to discharge fully, it could be difficult to recharge them again. Take the battery out of the strimmer and charge until it's full and then just top it up occasionally.