Dead battery sounds/then slow cranking/then car starts, in cold-ish weather

I suggest you have the charging system and battery checked out. The battery appears to have a low charge on it and this could be due to a problem with the charging system. Check the alternator connections for any signs of trouble and the belt tension. If you have a voltmeter check to see if the voltage at the battery is between 12.8 to 14.8 volts while the engine is running at around 2,000 RPM.