While the PC is shut down but plugged in the power supply is actually still powering the BIOS chip instead of the battery, so unplugging the PC starts draining this battery. You shouldn't worry much about this, the battery has a pretty long life (years) and it is replacable, and resetting the BIOS is simple.
If you are asking if you should disconnect your computer from power once the battery if full, the answer is no. If you do so, your battery will keep discharging and charging. The number of times batteries can be charges is limited and if you unplug your computer every time it is full, it will quickly die.
Should I either disconnect the power cable once the battery is full or let it plugged in while working? If you are asking if you should disconnect your computer from power once the battery if full, the answer is no. If you do so, your battery will keep discharging and charging.
The only reason to unplug it when you've turned it off is to help protect it from lighting strikes. And that's a very rare thing. Turning it off when not in use is probably a good idea in an area where power outages are common, though it's probably not really helping. It certainly won't hurt.
More of a topping-off when the voltage drops again. Some laptops permit operation with the battery removed from the battery bay. In addition to stopping charging, it prevents exposure to potentially damaging heat—heat is the main contributing factor to premature failure of laptop batteries.
From a view point of your battery's health you should keep the laptop plugged into the wall and unplug the battery from the laptop. Not because of over charging or whatever, but because of the heat generated by the laptop. By removing the battery it will be cooler, which is good for the battery.