You can get similar reactions with chlorine, fluorine, and some compounds of them such chlorine triflouride. Indeed ClF3 will combust with oxides such as sand and asbestos.

You can get combustion with various oxygen-containing compounds too, such as nitrous oxide.

A fire needs heat, fuel (something that burns), and an oxidizer. The most common oxidizer, by far, is oxygen, but several other elements can fill this role. Among others are fluorine, chlorine, and the other halogens.

Yes. In chemistry, the general term is "redox reaction" where one subastance is reduced while another is simultaneously oxidized. With fire, oxygen is reduced, while carbon, hydrogen, sulfer, etc are oxidized to produce compounds like CO, CO2, H2O, SO2, etc.

Other redox reactions that you might consider similar to burning without oxygen use other reactive oxidizers like Chlorine, Bromine, Fluorine, etc.

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