As French, I have no crisis of identity. I strongly believe in the values of the Republic and I am more a Soldier de l’An II than a Gauche Caviar. So I believe in the Ideal of Liberty, I wish we could have equality at least in the chance to start (but I am also aware of the Social Determinism) and I want to believe in Fraternity as a symbol, but also I took action to fulfil this last ideal.
My roots are the land where I born, in the mist and the swamp of the Dombes. My (family) history is a fight against oppression and struggle against poverty.
As I use to say we were Soldiers, Peasants and Workers. I am an incarnated Soviet.
I don’t believe in the France of the valets, the people who always co-opt with in inevitable and make money with it.

Now, you questions:
De Gaulle, not so much influence as such. De Gaulle was a man of the 19th Century, Captain during the 1st World War, and his values dated from this period. I don’t deny he was one of the men who restore the Honour of France, but he wasn’t alone.
I recognise his genius in politic, especially in 1940-1945 when he succeeded bluffing the Great Powers even without having cards in his hands.
He knew to surf on the waves, and was flexible enough to recognise when to change direction…
However, the Party allegedly heir of his ideas is just betraying all what, for what I know, De Gaulle fought and built…

1968 was a World Protest (it happened in every –almost- countries: Mexico, USA, Budapest etc) and was a bottle neck of Conservatism (in French meaning) and the access of the first benefits of modern society. To summarise I will mention Hippy Movement and Vietnam War.
In France it was the Post Colonialism and end of the War in Algeria, mixed with the Pro-Vietcong movement, an idealist vision of USSR, the apparition (the recognition) of sexuality and the possibility to have sexe without the pregnancy, the new economical model imported from U.S.A, the result of the Marshall Plan opposed to the leaders of a country who were still 19th Century moralists and a political Instability.
This and the demand of better salaries from the workers crated a explosive situation which created the Events.

I don’t know for the atmosphere itself, I don’t live in France.