Yeah, the HVAC requires very little vacuum energy. Is the sputtering sound pretty constant? If so, try this - remove the hose at the booster (that right-angle fitting is actually a check valve) and plug it with something, and start the truck. If the noise persists, the pump is suspect. If it goes away, then it's more likely the booster or perhaps just that check valve. You should be able to find that check valve in the Dorman/Motormite HELP section of your FLAPS.

Water pump pulley - look to see if it's just the pulley, or the shaft as well, that's wobbling. If the pulley is loose relative to the shaft, then it just might need to be tightened. If the whole shaft is wobbling, then the pump is probably on its way out.

And no, the tensioner should not move that much with just hand pressure on the belt. The spring is probably shot. Replacement tensioners have a longer arm. There's also the option of employing a SD tensioner, which also replaces the idler that's next to it. Requires a slight mod to the bracket. There's a Ford TSB about it floating around.