Do the aircraft report metrics such as fuel burn, ground speed, etc. continuously during flight to company ops/dispatch?

Yes.

They can and alot more, dependant on version/software/company preferences/costs.

Ours (at least 10 years ago) were sending lat / long, alt, TAS, fuel burn, fuel remaining, eta at next waypoint, and mantenance codes. I cant recall how often we had it set to transmit. I think we opted for each waypoint, but we could have it transmit every minute if we wanted.

ACARS only reports altitude, heading, air/ground speed and vert spd.

No, that's ADS-B.

It is really up to the company, ACARS transmissions can get expensive, it all software based. To my knowledge a variety of ACMS data can be transmitted if programmed to do so. We can also limit or expand on received data w&b etc.