Those two Futaba boxes should work together, yes. The best cord to use would be a Futaba brand cord with the correct type of connectors on the ends.

I've had several cases at the field recently where guys were using simulator cords as buddy cords. For some reason, this didn't always work, though I don't know exactly why, since I don't have access to the pin out information of either.

When you flip the trainer switch, the master should NOT have any control of the plane, the buddy box should now have full control. If the master still has control, that usually means that the master is not properly reading the buddy box for some reason.

No battery, antenna, or crystal is normal and fine for Futaba buddy boxes. The master provides power through the cable, and reads the control position signals. The buddy box does not transmit, and it's RF circuit won't even have power, so no crystal needed.

JR buddy boxes do require their own battery, since JR cables do not transmit power from the master to the buddy box like Futaba does. JR/Horizon makes a cable that is designed to allow a futaba buddy box to work with a JR master, but it doesn't work the other way around, and it's rather annoying to use, since it doesn't provide power, which means the Futaba buddy box would have to have a battery and be switched "on", which would then cause it to transmit if it had a crystal in it.