What is the code approved way to cut and patch fire rated drywall hung on the ceiling of a basement?
I'm not aware of anything special for fire rating a joint in drywall, just use the same joint compound and tape you would use on the rest of your walls. Fire rated drywall will cut similar to normal drywall, but it will offer a lot more resistance because of the embedded fiberglass.

When sealing any gaps or cracks, they make special fire rated expanding foam and caulk. They tend to be orange in color. There's also fire rated insulation that can be used for any utility chases/bulkheads that connect to the ceiling.

For junction boxes, we've been required to box our in on common walls (2x4 above and below the junction and a piece of plywood behind everything) to ensure that any shorts in the electrical couldn't travel to another unit, but I doubt this would be a requirement for anyone in a single family home.