crankshaft pulley

I don't think age is as big a factor in the exhaust manifold life expectancy. It is more so the mileage, heat that has been put to the manifold and any stress to the piece. Now how to check it.

I would take off the heat shield to get a visual observation of the manifold. Then unless something is suddenly obvious, I would start the vehicle and using a partner, raise the level of rpm to put additional pressure on the manifold. Using ears and touch you should be able to detect any leaks. To add extra pressure to the system, try temporarily doing a partial plugging of the tailpipe. You will be able to detect any leaks quickly.

Would a crack or leak in the manifold cause a decrease in the car's performance? My car is far from new considering it has 188,000 miles on it, but it doesn't seem to rev as easily as before.

You need to narrow down the location of the loud noise you describe. A drop in compression in the engine can cause significant power loss, but a vehicle that is not tuned, timed and running well will have power loss as well.

A good way to narrow down an exhaust leak is what it sounds like.

Leaks at the head or close to it (IE exhaust manifold gasket/crack in manifold) are often a sharp ticking sound, almost mechanical.

Buzzing like you describe is often a leak at the manifold/downpipe junction

the most common leak is between the downpipe and cat at the big doughnut gasket. this often manifests as a kind of puffing sound at idle.

did the pulley brak off or did the nuts come out.
if it broke off then the screws might have to be taped out. if they just fell out then get the right size bolts and reattach it. then put the belt back on

make sure you loosen the tensioner before you put it back on otherwise there is no way in hell you'll get the belt on....trust me i made that mistake and it was torture until i tried loosening the belt tension...rookie mistake :oops: