Hello! In this case, "mental skills" are not explicitly defined in the rules. A "mental skill" is anything the Keeper and players agree is a skill that comes from using your brain, not muscles or dexterity. Examples would be Accounting, Library Use, and History, but not Climb, Jump, or Fighting (Brawl).
My interpretation is that "minimum damage" is the lowest possible damage that could be inflicted by the source of the burn. So, for example, a Molotov Cocktail does 2D6 + burn. The initial damage is rolled on 2D6. Then, each following round the damage is 2 (1+1 = minimum possible roll on 2D6), then 4, then 8, until extinguished.
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