Why are diodes being used with these (apparently) AC relays?
Yes, the supply is being rectified just before each relay.  

Notice there is a diode in line with the controlling signal that is reversed to the flyback diode across each relay.

As the schematic on page 9 is part of a multiple page design, it is difficult to suggest anything other then to test the leads of the connector to see what type of signal is present when the switch in question is activated.

Based on your description, it may be that these signals are unfiltered rectified signals.  But this is a guess and should be tested.