The traditional location for the engine-chassis ground cable is under the car, from a bellhousing bolt to the chassis. You probably can't see it from above. But if you can start the car, you've got a connection somewhere. Still, if there isn't a heavy cable, you really should put one on.
Yes, you need clean connections at the generator mounting tabs, because that's the generator's ground connections. It's pretty hard to get this wrong, though.
The operation of the ignition light is pretty simple: it is connected from switched 12V (which also powers the ignition and almost everything else) to the generator output. So, the only thing that can turn it on is no generator output or some kind of miswiring. I suppose it's possible that the problem is in the box, but not likely. Make sure that the field connection (F at the box, small wire going to the generator) is OK. Make sure you haven't accidentally switched the field (F) and output (D) connections at the generator. This is hard to do, as they're different-sized tabs, but I suppose it's possible. Also check for voltage at the field terminal when the car is running. Maybe it's disconnected somehow in the control box or miswired.
My guess is that it will be one of those slap the forehead, geez-how-could-I-have-done-that kinds of things.