I don't know if you've checked and cleaned your battery terminals yet, but everything you're describing is consistent with corroded terminals. The dome light, instrument panel, etc. draw relatively little current, and will therefore work fine even if your battery terminals pretty badly corroded, because such a small amount of current draw won't cause very much voltage drop across a relatively small resistance at the terminals (c.f., Ohm's law), but the starter draws so much current that it will cause a huge voltage drop across even a small resistance at the terminals. (I.e., the voltage drop is proportional to the current draw.) In order for the starter to receive sufficient voltage to fire, your battery connections must be perfect.