Yes, the AC Filter capacitors in the output filter assembly are a likely culprit if they are old/original. Easy to check with basic tools/knowledge. If the inverter is actually damaged, you would need someone who can work on the unit to diagnose and make the repair.
The blast marks look like where a capacitor or IGBT exploded. Pretty normal damage when that happens right on the DC rail and since everything is so close together it doesn't take much debris to set it off. Likely whatever caused it was vaporized if it was debris. Ask for a failure analysis if...
The UPS output/inverter should synchronize to the internal/external bypass source when it is present. The internal/external bypass feeds must be from the same source. I don't know why the bypass would not be also coming from the emergency source.
Not to split hairs, but unplugging the UPS to test the battery isn't the best idea and could cause an issue. In the future just open the circuit breaker that feeds the outlet.
