Difference between a DC SSR (solid state relay) and AC SSR
What would happen if one accidentally applied a DC load to a VAC solid
state relay and vice versa?

An SSR (solid state relay) that uses a triac as the switching element, when fed from a DC supply to a load that takes DC current, will never turn-off once activated. On an AC supply, by virtue of the supply alternating, it passes through zero every few milliseconds and, commutates the triac into an off state when it is no longer being activated by the control signal.
A DC SSR might not be capable of dealing with the reverse voltage situation of AC because it might only use a single MOSFET as the switch. What would happen then is that when the supply reverses, the MOSFET will act like a conducting diode hence, it would be inappropriate to use it in many situations.

Can one convert between each type with the addition of simple
components (e.g. diodes, optocouplers, etc.)?

No.

If I have a completely unmarked relay, how can I determine if it is
for VAC or DC loads, preferably using a multimeter or similar device?.

The same with any device that has no provenance; it's pretty useless and unreliable in all but the most desperate of situations. Don't get into that desperate situation is my advice. If you need an SSR, buy one that has provenance and a data sheet and one that is sourced via a reliable distributor from a recognized original manufacturing facility having decent quality standards.
What to check for when buying an electronic component or module.