Q: Leakage near the radiator fill cap and the AC/Heater doesn't work

It sounds like you have 4 issues:

1. Coolant leak

2. Possible issue with the water pump

3. An unexplained, but transitory noise

4. HVAC faults.

The coolant leak is the most important one to fix first because a coolant leak can eventually cause overheating which will basically destroy an engine by warping metal parts so I encourage you to resolve the leak promptly. The water pump issue would have to be "explained". Not sure what your Mechanic saw or suspected but if there was a problem it should have been stated to you with specificity (e.g., pump is leaking; pump is noisy; pump drive belt is loose; pump flow is inadequate).

YourMechanic can take the general observation you received and, after inspection, put the diagnosis in concrete terms for you (if the pump is even bad). Momentary engine noises, at start-up, can be due to initial lack of oil pressure, belt pulleys (bearing issues), or like problems. These are relatively easy to evaluate on inspection, particularly if they are momentary. And, the serpentine belt can be temporarily removed thus quickly distinguishing between "engine" noise and noise from the accessory devices driven by the belt.

The AC system and the heater are distinct systems for the most part, excepting they share common air vent pathways. This is the last thing to fix though. The leak, the water pump and the noise issue should all be resolved before you put any money into HVAC repairs (in the olden days, cars did not even have HVAC systems).

To avoid any unnecessary repairs, YourMechanic would be pleased to diagnose your coolant leak issue for you and address any further concerns you might have. Please let us know how we can assist you!