Hard-core evidence? I don't recall seeing anything set in stone, but I think that you can assume that there are birth certificates of some sort because of how they track underage magic.
First, do you think or is there any canon evidence as to official birth certificates?
If underage magic flares up somewhere and there are no registered wizards there one could just assume that a wizarding family had gone on vacation, and that someone's kid had gotten a hold of daddy's wand. But if underage magic continually flares up there and there is no record of someone magical living there there is no way for you to send that person warnings, call them to hearings, or arrest them.
There probably is, seeing as the ministry has tabs on where family's of wizards and witches are living. This is so that if any family member performs magic in front of a muggle the ministry can track down said family.
What about records of marriages?
I suppose it would depend if the marriage done in secret was done with any magic. If the marriage was done the muggle way with no magic involved I would assume that the ministry doesn't know.
Would the Ministry have records of all marriages? What about ones done secretly with only a few people present?
I would think that there isn't a branch devoted entirely to weddings, but maybe a branch devoted to social events and gatherings. Remember, the same wizard that conducted Dumbledore's funeral also conducted Bill and Fleur's wedding.
Also, is there an official branch of Wizards who perform weddings?
I answered all questions from the view of the Ministry doesn't have magical trackers on every witch and wizard.