Is my External display killing my MacBook Pro battery?

Do i really have to unplug the Macbook after every time I stop using it?

Not exactly. Unplugging a portable Mac only when it is not in use is not the right thing to do either. Batteries need to be exercised routinely, or they will suffer a premature death. Like people.

macOS 13.2 is significantly outdated and does not incorporate recent power management algorithms designed to maximize a Mac battery's longevity: About battery health management in Mac laptops - Apple Support. If you leave your Mac connected to a power source indefinitely it will definitely reduce its useful life.

To learn how to update macOS please read How to update the software on your Mac - Apple Support. Have Apple replace the battery. The Mac won't work properly until they accomplish that.

It's not necessary nor is it beneficial to fully drain the battery once a week. It just needs to be used regularly. Using the Mac on battery power alone for a few hours once in a while is sufficient. Exactly how often and for just how many hours are not very important, but the opposite extremes (never using the battery, or leaving it in a discharged state for many weeks) are certainly detrimental.

Apple used to provide suggestions along those lines but recent macOS improvements in battery health management technology have made them obsolete. Lacking those suggestions, all I can recommend is "just be sure to use it once in a while."

Any extra load you put on your battery will hasten degradation, but the incremental amount is likely minuscule. Using an external monitor is not damaging to your battery, it just may a shorter run time.

Longevity is a more complicated thing in reality. Batteries are made to be used and recharged but that very action is "using them up" over the long haul. That's just the way batteries work. Some fail earlier than others, some last for years and you can't predict which will happen.

Just use the monitor, charge the battery when needed, use on battery when you need to and let the battery circuitry handle the details.