Dandruff shampoo is a little more heavy-duty than normal shampoo; people with dandruff have excess sebum that causes the skin to flake off, necessitating a stronger formula.
You can even sit under a heated dryer to increase the effectiveness of the conditioner.
You will not need to re-color your hair; the paste is non-damaging.
Most dyes are meant to handle alkaline substances, like soaps and shampoos, but not acidic substances. The acidity of white vinegar will help to remove the dye.