The life span of the battery is not affected while using the battery packs on mains.
You can leave the charger connected, but if you shot with high flash power in short cycles, the battery will discharge even with the charger connected.

If you do not use full power flashes, the unit can take a breath after a series of flashes, and the unit will not overheat. Don’t place the unit in the sun or close to heating systems, and take care that there is air circulation, not using the unit in a bag or similar.