A It should be: I had my doubts about its happening.
"About its happening" is a prepositional phrase. The object of the preposition is "happening." "Happening" is a verb form used here as a noun. (-Ing words used as nouns are called gerunds.) "Its" functions as a possessive that modifies "happening." Pronouns or nouns that precede gerunds need to be in the possessive form. If we change the pronoun "its" to a noun, the noun would have to be possessive: I had my doubts about the event's happening.(not event happening)
Notice that the possessive pronoun (its) does not have an apostrophe, but the possessive noun (event's) has an apostrophe.