Fire basically conducts electricity the same way salty water does: both contain some concentration of charged particles that are free to move.
Water contains some concentrations of ions and protons (H++ protons). When there is a voltage difference, the ions will move according to their charge.
The hot gas of the flame contains positively charged ions and electrons, which will move in the same way as the ions in the water. The reason there are ions is that the heat of the gas is such that some of the electrons can free themselves from the attraction of their atoms.
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For sure! Fire contains some plasma, and plasmas have freely-moving charged particles within them. Anything that has freely-moving charged particles conducts electricity.
So metals, which have a sea of mobile negatively-charged electrons on their surface, conduct electricity.
