Never heard of anyone doing that before as for the laptops as far as I knew ran off of the secondary circuit or at least used to. Primary circuit being the charging for the battery, then the secondary being the laptop drawing off of the battery that is powered through the primary.
If your laptop does run without the battery attached it is VERY dangerous to your data on your hard drive. The slightest power disruption can crash your laptop without the battery acting as a reserve.
I personally would just make a habit out of plugging in the laptop when using it and outlet available, and unplugging it when not in use so the battery does not build memory and go bad on you.
Also overheating of a battery is a good indication on that old laptop that the cells in that battery had had it and a new battery probably would have operated cooler. When batteries start to go back they resist taking a charge and generally produce heat by this resistance to take the full charge.