On a lead-acid battery, neither set of plates should be grounded unless it is installed in a system that grounds one side, as in most automotive and generator starting systems.
Starting batteries often have a rough life. Our company SOP calls for replacement every three years unless problems surface earlier. Trying to push past three years is asking for trouble.
I agree that flooded-cell lead-acid batteries in many services can last ten-twenty years with proper care, but that's a different game.
Following a start attempt, it is not unusual for the battery to make some heat as it recharges. It should not "hum" unless the charger is introducing an AC component into the charging circuit.
One problem not uncommonly found with starting battery sets is that some manufacturers aren't reluctant to sell a multiple million dollar generator installation and then install a cheesy battery charger along with it.
Refer to battery manufacturer literature for the proper charging parameters and make sure your charger is supplying proper voltage and that it is reasonably free of ripple. If the voltage is wrong and cannot be adjusted, consider installing a better battery charger.
old field guy