No, the word 'of' is not a noun, not a possessive noun. The word 'of' is a preposition, a word that indicates origin, relationship, cause, or measure. Examples:
I tore the sleeve of the jacket.
The first of May is her birthday.
They are two of a kind.
The possessive singular noun is explorer's. The possessive plural noun is explorers'.
The possessive form for the noun coyote is coyote's.
The possessive form of zoo is zoo's.
The possessive form for the noun preacher is preacher's.
The possessive form for the noun laboratory is laboratory's.
No, he is not possessive. The possessive form would be his.
The possessive form for the noun oxygen is oxygen's.
The possessive form is grandson's.
The possessive form is island's.
The possessive form for the noun coyote is coyote's.
The singular possessive noun is Rex's.
No, it is a singular possessive noun.