No, Latin was and remained the main language of philosophy, science, courts, the Court, and the Church. It was the language of the educated elite; the vulgar Latin of the common people gradually morphed into the Romance languages.

French grew increasingly important throughout the Middle Ages - it was the language of the English ruling class for about two hundred years after the Norman Conquest - and eventually supplanted Latin as the 'lingua franca', but putting an exact date on that change is probably impossible. The treaty of Westphalia in the mid-17th century was, I believe, the first to be written in French instead of Latin and is probably the best indicator of French's ascension.

So if you're asking for 'where', the answer is 'at least all of Europe', and 'how long', then 'about 1200 years after Odoacer killed Romulus'.

I'm not sure what you mean about 'tracing the linguistic influence regionally.'