Yes, a laptop uses battery power when it is plugged in. The amount of power used depends on the model of the laptop and the settings chosen. For example, if the laptop is set to run at full performance all the time, it will use more power than if it is set to run at a lower performance setting.
The laptop uses ac adaptor when plugged in. Not battery. If you take a look at the battery icon when plugged in, you would see it goes to 100, then gets discharged to some level and charging back again. So the simple answer is that laptop does not use power from battery when plugged in. Battery gets charged and discharged. Does this kill battery?
The same for your laptop, if you have removed the battery. The battery works like a UPS (uninterruptible power supply), so you can still use your laptop and turn it off safely. I'd keep it plugged in. Some laptops have a threshold where the AC doesn't charge the battery at all until it drops below a certain point.
Most modern computers have the capacity to manage this scenario. Once a laptop battery is fully charged and the laptop is used while plugged in, the battery is usually bypassed for power consumption, meaning the laptop runs directly on power from the wall outlet. This means it should not harm the battery directly in terms of charge cycles.
A fair number of laptop makers include a power management tool to protect battery health when plugged in for days at a time--Lenovo (Thinkpad series), Dell XPS/G series, Alienware and Acer have charging limits so the battery will need to be drained to 80% before it fully charges and counts as a charge cycle.
Yes, you can use your computer plugged in even when batter is full. Most modern computers have the capacity to manage this scenario. Once a laptop battery is fully charged and the laptop is used while plugged in, the battery is usually bypassed for power consumption, meaning the laptop runs directly on power from the wall outlet.