The 5V0 is simply the 0v reference for the 5V supply. Similarly the 24V0 for the 24V supply. You *can* connect these together - it just means that the supplies then share a common reference.

The isolated outputs from the AXBB are (I said Open Collector, from recent posts it's clear these are Open Drain - difference in transistor technologies - FETs vs Bipolar, but ignore that). This does mean though that the output FET Source is connected, in the case of the AXBB to the 24V0 supply. Therefore, to switch the digital input to the VFD (which is to provide a low-impedance path between VFD DCM and FOR/REV) then you need to connect the FET's source (24V0) to the DCM line on the VFD and the drain (O2/O3) to the FOR/REV input on the VFD, or it simply won't work.

One other thing to try, to help diagnose your problem - you're 3-core CY cable - with the DCM/FOR/REV wires - just try shorting the other end so DCM & FOR are connected - you should see the VFD display change (from memory it goes from a flashing display to a solid 00000 to indicate a demand for 0RPM... but that's just a distant memory).