An AC SSR normally uses a triac. When you release the gate voltage, the triac will continue conducting until the voltage across the triac (and the current through it) go to zero.
A DC SSR will use a MOSFET, and this will not provide the latch function of the triac. Furthermore, the MOSTET will only block current in one direction - in the other direction the body diode will conduct, so the SSR will conduct half the time even with no gate drive.
"What would happen if one accidentally applied a DC load to a VAC solid state relay and vice versa?"
If you use an AC supply (not load) with a DC SSR, you'll get power to the load half the time, with the other half of the time being either on or off depending on gate drive.
If you use a DC supply with an AC SSR, you're hosed. Once you've turned the load on, removing the gate drive will not turn off the load - and it will remain on until you turn off the DC supply (or remove the load from the SSR).
"Can one convert between each type with the addition of simple components (e.g. diodes, optocouplers, etc.)?"
Sort of. Sometimes.
If you put AC through a full-wave bridge and apply that to a DC SSR, that will work.
There is no easy workaround for a DC voltage and an AC SSR.