Can I use the charger to power the computer without the battery? Will it damage the computer?
Every laptop I have owned to retirement would work with the battery removed and/or with a completely exhausted battery installed. IMHO, most laptops are more ergonomic with the battery installed.
On an aside, it is best to avoid intentionally draining the battery. Li Ion batteries are good for about 300 charging cycles. So unless you need to use it off the cord, save those cycles.