If battery will not hold a charge, it will eventually not turn the engine over if it sits long enough. I would say that before replacement the alternator output needs testing with it on the car (don't let ANY idiot disconnect your battery with the car running "to test the alternator", sure sign of someone that doesn't understand how the alternator works) and if it checks out, replace the battery.
Batteries fail of their own accord, a modern "good" battery, taken care of properly, is lucky to go 5-7 years.
With all the systems that draw power from the battery in today's cars when the car just sits eventually they will drain the battery. In normal everyday use this isn't a problem. In fact a good battery will hold a charge for a month or more even with all the demand on it. The alternator does not hold a charge. It's function is to recharge the battery while the engine is running. If your battery won't hold a charge then your battery is bad and you need a new one.
YES, it will discharge to a point of not cranking over.
YES, the alternator recharges the battery and also powers all the accessories whenever they are on.
The battery holds the charge, not the alternator. When it doesn't it needs to be replaced and the system (both battery and alternator) tested.
if battery not holding charge and it's not old, alternator is probably bad - you need both to be good
If your battery isn't holding charge, will the car eventually lose power? Or does the alternator charge the car when driving? However, if you don't have a good battery, how does the alternator hold charge?