You need a hydrometer to accurately tell if the battery is charged. However, 12.6 is generally a fully charged battery. These readings should be made about an hour after it has come off a charger so it can 'calm down'
12v on a battery means the battery is very low. If a battery is ever below 10.5 its beginning to be damaged.
A single light on for a week is not gonna kill a battery as at worst its about 1 amp of draw. (Unless its a lot of lights)
It is normal for a charger fan to cycle on and off. Its not however normal for the voltage to cycle up and down as rapidly as you said it was.
What type of battery did you buy? Is it a flooded RV type, or an AGM, or a GEL.
Batteries will get warm while charging, but not hot. They will also fizz a bit and very likely smell like rotten eggs. This is esp. true of 'true' deep cycle batteries.