If you're using the laptop as a desktop (it has a permanent spot on your desk or kitchentable for example) like 90% of the population* then you actually don't even need a battery. Have it plugged in at all times. To answer question 2, yeah it's perfectly fine and your laptop don't mind it at all.
If you get a new battery keep in mind that if you know you're going to have the laptop on the same spot for quite some time then it's best to remove the battery until you need to move the laptop. The battery is constantly drained and recharged if the laptop is plugged in.
*made up numbers based on experience. The pleb actually believes that desktops are ancient machines lol.