Add water to what and make acid? Wood ash is a source of alkali if anything. The carbonized content is difficult to completely get out of any rug as it is not soluble, though mechanical flushing can get some of it.
Oh, it’s a good idea that when the rug has dried after shampooing, apply a Scotchgard type spray.
Correct.
Wood ash is mostly mineral oxides, which, when mixed with water, make alkali solutions.
That’s how they make soap.