Dual conversion only refers to the filtering the receiver does on the signal INTERNALLY. It has nothing to do with the signal between the transmitter and receiver, which is standard and hasn't changed in decades. There is no such thing as a "dual conversion transmitter" for dual conversion receivers.

Your old FM radio will work fine with any JR-compatible FM receiver, single or dual conversion.

ALWAYS use the crystal that is specified by the manufacturer for that receiver, and only that crystal. That's the crystal it was designed to work with, and any other crystal will most likely offer degraded performance, if it works at all. There's no need to run a JR crystal in the receiver just because you have a JR transmitter. The receiver understands what the JR is saying just fine with its own crystal.