Several constructions exist that link two nouns with the preposition de.

1. The preposition de can be used to characterize the kind of object we are talking about. It is in this case used without an article, and the following characterization can either be singular or plural.

   le sac de sable
   le sac de fraises

   This is what happens with “justificatif”.

   le justificatif de domicile

   You can think of it as a single word, like “brosse à dents”.

2. Other usages of de exist, for example, to express possession (but not only). In this case:

   • to form an indefinite reference use d’un/d’une (preposition de + article) + noun

     le sac d’un ami

   • to form a partitive (or indefinite plural) reference use de (preposition de, alone) + noun

     plusieurs sacs d’amis

   • to form a singular definite reference to an object use du/de la/de l’ (preposition de + article) + noun.

     le sac du facteur

   • to form a plural definite reference to particular objects use des (preposition de + les) + noun (plural form).

     le sac des élèves

The funny thing is that many of your examples (“beaucoup”, “peu”, “nombre”) are very particular examples, because each corresponds to two different fixed locutions. The first ones are very similar to first type of constructions:

• beaucoup de bateaux: a lot of boats
• peu d’élèves: few students
• nombre d’instituteurs: numerous teachers

The second ones act as a particular kinds of pronouns and are quite similar to the second type of constructions as they refer to a definite set:

• beaucoup des bateaux (que …): many of/among the boats (which …)
• peu des élèves (que …): few of/among the students (whom …)
• nombre des candidats (que …): a large number of/among candidates (whom …)

It is also possible to use un/le nombre as a usual noun. In this case it has to be preceded by an article and it means “number”. It is followed by a bare de:

• le nombre de participants: the number of participants

It’s similar for un/le peu which means “a/the little” (and can also be used for some uncountable plurals):

• un peu de sable
• un peu de fraises