Cold affects the speed at which molecules are moving around. Fire is a phenomenon produced generally by exothermic chemical reactions between molecules. The part of the fire you see is, if I understand correctly, the photons emitted by excited electrons moving to a lower energy state after the reaction.

All other things being equal, the colder it is, the less the candidate molecules are moving, and thus the less likely they are to encounter each other and react. IIUC almost all chemical reactions are less vigorous at lower temperatures, disallowing convection of course.

At absolute zero, ambient oxygen would solidify. Not sure what happens if you drop a match on a bunch of solid oxygen, but I imagine it's not good.

It would go out. In a room cold enough for Oxygen to turn solid, there would be no gaseous oxygen to sustain the match’s combustion as it falls. The wood would have some latent heat, but not enough to re-ignite itself in a room that cold.

Pure oxygen won't burn anyways. What would it oxidize? If you had frozen a mix of oxygen and hydrogen (without the two separating) it would be possible to ignite and burn it.

It would oxidize the match that’s being dropped in.

Truly pure oxygen won't burn, it needs fuel. So, what would probably happen is it would burn super bright for a split second as the match itself was consumed, and then extinguish itself as it has no more fuel.