It can do far worse, such as destroy your engine block. I wouldn't worry about your heater core, I would worry about your Engine.
Yes. The coolant carries the engine heat to the heater core. Without coolant the heater core does not get warm.
A leaking heater core will cause the engine to overheat because the coolant is leaking out.
Only if the coolant is leaking from the heater core, but you will see this.
If your heater core is LEAKING engine coolant so that your engine coolant level drops too low or the engine coolant is lost completely and causes your engine to overheat then your engine can be damaged beside the mess the leaking coolant makes inside your vehicle ( you might be able to just temporarily bypass the heater core so that no engine coolant is flowing through the heater core )
You may smell coolant when the ac is running but a leak in the heater core won't cause the ac not to work.
No it wont. The heater core is just a small radiator inside the cabin that when the fan is turned on heats the air. A bad heater core will cause you to have no heat inside or will leak coolant on the carpet.
Yes, if the coolant level is low, or the pump is not capable of moving coolant through the heater core.Yes, if the coolant level is low, or the pump is not capable of moving coolant through the heater core.
The presence of the valves usually restrict the coolant flow into the heater core.
Heater Core
Leak in heater core
Sounds to me like the heater core is leaking coolant.
A damaged heater core will cause coolant or water to leak onto the passenger floorboard.