Charging problem is likely either generator or regulator. Both are still available thru different sources. With the number of hot rod 57/58 Plymouth and dodges being built on this site you might even be able to find good used units laying around someones garage. An alternative to replacement is also rebuilding the generator (not a hard job with a service manual handy) and cleaning and gaping the points on the regulator. An alternator is also an easy upgrade......if you go that route, I'd recommend using an alternator with an internal regulator or a 1 wire unit.
Because of the design of the heater plenum on the outside of the fire wall and the drain holes located below the core a leaking heater core generally does not leak inside the car. The most likely source of the leak inside the car is the heater control valve (where the heater hoses connect on the lower right hand side of the heater box on the firewall. Rebuild kits for the valve used to be available from NAPA but don't know if they still are. A universal heater valve can be used in place of the original (spliced into the heater hose) but will likely require changing the control cable on the lever.
Check to make sure both screws that hold the horn ring are there and tight. The correct screws will bottom out at the correct depth to allow the horn to be pressed but not make contact until the horn is pressed. Also make sure the spring is in pl,ace under the center of the horn ring.
1957 Plymouth (Hemi, Dual Quads, A833 4 Speed 9 1/4 w 4.10)
1937 Dodge Pickup (Hemi, 6X2 intake, 46RH, Dana 60 w 4.56)