How could I produce/rig this flame-colour lottery?
Smokeless and quick burning isn't hard to achieve, just use flash paper. For white smoke, I would use flash paper again but with a white oxide powder such as titanium oxide with a flame retardant to slow the burn. For red, again flash paper but, you need a compound that releases bromine when burned. I think copper(i) bromide or silver(i) bromide would be ideal as both are easy to decompose and tolerant to moisture in air, though silver bromide is not tolerant to prolonged light exposure. These salts may discolor the paper and you may need to color the non-red smoke papers to make them all look the same.