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You need a hitch that fits your frame width. I put a Valley on. Class IV actually. Now, no one will list 61-66 becuase there were changes in that era.
Just get one that fits a 67-71. You will need to drill some new holes though the frame, from underneath. At least with the Valley. But think you will have to do that with just about everyones. I cut out the frame rivets holding the rear stiffener (rail that goes between the frame sides) and then used those holes, plus a new hole further back. This makes for a helluva good attachment to the frame. My hitch is way stronger than the rest of the vehicle can take, in terms of load.
No it will not interfere with a between the rails fuel tank, I think. The mustang tanks sit between the rails.
However, it WILL interfere with your spare tire, especially if you have a SWB.
No idea about dual exhaust. My six banger exits in front of the rear passenger tire.
Mine came with a Valley on it. Mine is a long bed, no issues with the spare wheel. It is the same width as the frame rails, so running duals was not a problem.
One thing I do suggest is stripping it and getting it powder coated. Mine was getting extremely rusty...seem like the factory paint on it didn't adhere worth a darn. Better to do it now than later.
I have a local dealer installed class 3 hitch, and had the duals added afterwards. There is no standard placement for the pipes and they can be cut and moved if need be.