It seems the battery goes dead quicker if I turn the phone off, why is that? I never turn it off now.
It's a well documented problem on the boards. I haven't seen any definitive answer as to why exactly it happens (usually something to do with certain Android processes still running differently while off, yada yada yada), but it does seem to be fairly common. You already found the easiest/most suggested fix for it... not turning the phone off. There's no worry of overcharging these batteries, so just leaving it plugged in overnight, every night is no problem.