Doesn't necessarily sound like the alternator to me. Sounds like it could be the fuse box feeds on the positive battery terminal have corroded or are damaged causing a significant voltage drop. Common problem. The solution is to cut out the "fuseable links" (the first 10 or so inches of the wires to the joint you'll find if you undo the wrapping/sleeving) and replace with the appropriate gauge of new wire.

This could potentially cause all of the above faults including the missfire if the ECU voltage is too low.