"Do I need to replace the breaker box?"

Yes, ASAP, with a proper ground.

That's definitely an odd one. First things first, turn off the feeder to the subpanel, at the other end. You definitely have an unsafe condition.

After the power is off, check that all the neutral connections are tight. If one is absurdly loose, that may be your issue.

If that is NOT your issue, then please, DO NOT turn that panel back on until the true problem has been found and solved.

It may very well be that the panel itself is structurally damaged - although I'm having a tough time wrapping my head around how that would lead to sparks from the NEUTRAL bar.

I try not to give this answer, because it seems like a cop-out from helping people learn, but this is a pretty serious issue, and you really do need to call in someone who knows what they're doing. I DO believe in DIY, as long as there are forums like this to help, but when sparks and smoke are already involved, things become time-critical. Whatever the problem is could very well set the house on fire while you're waiting for us to reply!

So get that feeder turned off. NOW. Don't wait.

When would you consider calling a professional in, after the fire dept. has left?