- Where/how is the MagSafe port connected to the rest of the machine?
It is on the right side of the board if you flip the MacBook air upside down.
- If so, which connection is likely broken, the port to board, or battery to board?
Honestly it's probably a chip or a few other components on the charging circuit that has gone bad.
Does the MagSafe charger show any lights right now? This info is important as it can tell you what could be at fault.