Unless you are playing games, you generally do not need a cooling pad. If you think it runs hot just in general usage, then sure a cooling pad could help.
When, plugged in, it is always a good idea to remove the battery, unless you are charging it. When the battery is fully charged, then remove it. Overcharging a battery can lower decrease lifespan even though there is circuitry to prevent overcharge.
Most people traveling around with a laptop also have the battery attached. Can't think of a reason why it should be removed.