Is my laptop broken or is it bad for the battery?

I play gta on my laptop and I always leave the battery cable inside because otherwise it quickly goes empty, which means that my battery is always at 100 percent, does it damage the battery or is it not bad?

No, your laptop is like a stand PC. Useful when playing. Laptops reduce performance to battery mode. The battery can't get fuller than full, I do it because I use my laptop for gaming.