"It is my wedding."
Translation:Es ist meine Hochzeit.
Hochzeit is a feminine noun, the correct article for it is die in the nominative singular. German pronouns match the gender of the word they replace. Therefore I would expect it to be replaced by the third-person singular feminine pronoun, not the third-person singular neuter pronoun.
Yes, you use the appropriate (er/sie/es) when the noun has already been mentioned.
e.g. Die Hochzeit war sehr schön. Sie hat mir gefallen = "The wedding was very beautiful. I liked it."
But no, you use es when referring to something before the noun has been said or otherwise clarified by context.
e.g. Viele Autos sind vor der Kirche. Es ist vielleicht eine Hochzeit = "Many cars are in front of the church. It might be a wedding."
Not quite. The 'institution' is called 'die Ehe'. "Heirat", "Trauung" or "Eheschließung" mean the act of marriage. "Hochzeit" can also be used in that sense but its main meaning is that of a wedding celebration. "Heirat" is often used for the act of marriage in a legal sense, not for the church ceremony. The entirety of the ceremony in church is called "Hochzeitsmesse".
To answer the original question: Nobody says "meine Heirat" to refer to their wedding. When you say "my wedding" you probably mean the whole ceremony and the party afterwards. The best fit for that is "Meine Hochzeit".