An abbreviation is a truncation of a word, such as the use of inf. to mean infinitive. Fais and ai are conjugations of the verbs faire and avoir, not abbreviations. The infinitive will be the verb form that appears in the dictionary, and it must be conjugated before it is directly applied to a subject. To make an imperfect comparison to English grammar, is and am and were are conjugated forms of (to) be in English, rather than abbreviations.
As implied by the above, je avoir and j’avoir are ungrammatical. Avoir is the infinitive of the verb meaning "to have." The conjugation for the first person present is ai. Because ai begins with a vowel, there is elision (not abbreviation) in je + ai, so we say j’ai.
For a more comprehensive introduction to verbs, please refer to a guide such as Tex’s French Grammar.