So there is a ~3/4" pipe that comes off the exhaust manifold, it goes towards the front, has a coil for cooling, goes through what looks like an air conditioning dryer, then a hose that goes to the air cleaner. I assume this is some kind of emission devise. What does it do and do I need it? I would like to use a smaller air cleaner (Weber) and need to know if I need to keep this thing.

Sounds like you got some smog stuff that was half removed , like the smog pump and hand full of vacuum lines ,
Post some photos of what you have now , a picture is a thousand words 🔩

Are you in California ? What year will the car be registered as ? Pre 1974 get rid of it! After 1975 it's part of the Thermactor system for emissions. This system was used on most Ford cars until the EEC-IV came out. So if you have to have it you should be able to get parts. That dryer as you called is an air bypass valve or the air bypass valve. As per the Ford Car Shop Manual 1975-1976.