Measure your chain. It should measure exactly 12 inches from outer plate to corresponding spot on the 12th link away. If it is more than 12 1/8", then it's toast and probably the reason you broke a tooth. If the chain is NOT worn, it could still have sand in it to make the grinding sounds. Or if by "grinding" you mean "clattering," then you just need to adjust the cable tension so that the jockey pulleys line up with the gears on the new cassette.

You mention the outer two gears not being riveted. This implies to me that you have a cassette on a freehub, not a freewheel.