I believe the twist here is that:
Dominique was born on Dec 31st, or Justine was born on Jan 1st.
Why do we need the twist?
It's not too hard to rule out everyone else as being Alexandre's wife. Nicolas is definitely a man, and Claude and Frederic (likely men anyway) were born on Fridays. Noelle being older than her husband doesn't fit with being born in the same year, both on Mondays. Clara being born in 1982 means if she was Alexandre's wife they'd have had to be married in a leap year and he couldn't have been born in January.
Consider Dominique being born on Dec 31st. Then her wedding was on a Thursday, January 1st. Alexandre's birthday could be any Friday in January or February, and then Dominique's birthday would be Thursday, Dec 31st of that year. This requires that they were born on a leap year and married in a non-leap year. Because Clara was born in 1982, Dominique and Alexandre must've been born in either 1980 or 1984 to satisfy the ages not varying greatly. Dec 31st 1980 was a Wednesday, but Dec 31st, 1984 was a Monday. Being born in 1984, they could have married in either 2009 or 2015, as those are non-leap years with Dec 31st falling on a Thursday.
Now consider Justine being born on Jan 1st. Then her wedding was Friday, Dec 31st, and her birthday was Saturday Jan 1st of the next year. So Alexandre and Justine must have been married on a leap year, and Justine's birthday in the year of their wedding was Thursday Jan 1st. However, if you look at Jan 1sts between 1978 and 1986 you'll find that only 1979 had Jan 1st as a Monday. They could have then have been married in 2004, which is a leap year in which Jan 1st was a Thursday.
Now that we have some possibilities established for that person, let's move on to another one.
Let's figure out who Nicolas' wife is. His wife was born on a Friday 13th, so Noelle, Justine, and Dominique are out. The last three people are Claude, Frederic, and Clara. Clue five appears to state that three people were born on three different Friday 13ths in a single year. However, in 1982, when Clara was born, there was only one Friday 13th. So the clue should instead be interpreted as saying "Claude, who is Nicolas' wife, and Frederick..." This results in the women being Noelle, Justine, Clara, and Claude, which also explains why the wording in clue four is so careful to not use a pronoun for Dominique.
We now know who is married to whom:
Alexandre is married to Justine. They were born in 1979, and married on December 31st 2004. Nicolas is married to Claude. Clara married Dominique on January 6th, 2005 - the only Friday, Jan 13ths near 1979 were in 1978 and 1984, so to be born in January Frederic would either need to be about 5 years younger (not classmates) or the oldest. This leaves Frederic and Noelle being married.
As for the May 12th, 2005 wedding
We don't yet know when Nicolas and Claude or Frederic and Noelle were married. From Wikipedia we can see that when May 13th is a Friday it is the only one in the year, so neither Claude nor Frederic were born in May. This leaves Nicolas as the only possibility, so Nicolas and Claude were married on May 12th, 2005 before Nicolas' birthday the next day.
When was Clara's wedding?
January 6th, 2005