Buying of books/selling of books
What article should I use before buying/selling? All examples I saw on the internet have "the" before buying/selling. Why?
They are preceded by the definite article as they basically become a noun. "Ing" in this case is used as a suffix to form a noun.
To do the washing -
This is an example of a deverbal noun, or the progressive form of a verb being used as a noun.
See: https://en.wikipedia.org/wiki/Gerund#Roles_of_%22gerund%22_clauses_in_a_sentence -- in particular, the sentence, "Her playing of the Bach fugues was inspiring." In general, when you have a noun phrase of the form "VERB-ing of X," you need some kind of article or possessive pronoun before it. I am not sure if there is a particular name for this rule, but it sounds incorrect to omit the article (or possessive pronoun).