Is it good if you have the plug of the charging cable permanently connected to the laptop?
It is not good for the battery when it is constantly charged to 100%.
If no battery is in it then yes.
No, that's not good for the battery! I speak from experience, because that's why one of you almost burned down. They bloat after a while when they are permanently overloaded and then the fire danger is very high. Take the battery out if you use your laptop as a tower.