The French are largely of Celtic descent, with some Frankish admixture in some regions, in others, practically none. Except in Gascony and the Pyrenees where they're Basque in descent, Gascon being of the same etymology as Basque.
People of Frankish descent are mainly found in the Rhinelands, which does include a bit of France, but is mostly located in modern Germany.
Yes - The Celts were the people spreading throughout Europe during the late Bronze Age and the Iron Age. They spoke the Celtic language and shared cultural and religious practices and beliefs. The Gauls were the Celtic tribes that settled in what the Romans called Gaul.
Sorta kinda; "Celt"/"Celtic" is a broad term that's been applied to a lot of different peoples, languages, and cultures. The history blog ACOUP has a good post on it [1]; it's long, but only the first part is really relevant to what you're asking (up through the table comparing different regions).
It's an exonym, i.e. it's not how the celts living in Gaul would call themselves.
There are two origins of the word that are likely to cave coexisted in a "convergent etymology"
1. The name of an ancient Celtic tribe (one of many) sounded like "galatai" to ancient Greeks and it sounded about right because of the the white skin (either fair skin or white paint). So in this case it ultimately stems from a Celtic word.
2. The old Germanic word "walhaz" meant something like "foreign" or "Roman". That originated many names for former roman territories when Germanic people encountered them. (Similar word for Slavic). That left a lot of names like "Wales/welch", "Wallachia", "vlachi". This word went through a sound change when adopted in romance languages similar to how other words like "werra" - > "g(u)err(a|e)", "Ward" - "g(u)ard" ...
The regular sound changes in languages betray the coexistence of these two etymologies at different times and phases of the Evolution of the romance languages and french in particular.
The latin "g" sound evolved into "j" (gamba -> jambe) so one our think that if the word "Gallia" was available from the latin substrate it would have transformed into "Jallia" and in fact it was! As attested by place names like La Jaille-Yvon and Saint-Mars-la-Jaille.
This reveals how the modern french name "gaule" likely stems from the second etymology (the German origin).
That said, languages are rarely let evolve naturally. There is plenty of people who studied languages and kept latin alive for centuries and it's also quite possible that the version with the "g" sound has been also kept reinforced because of the relationship with the classic texts (Gallia est omnis divisa in partes tres)
The people most likely to threaten your borders are also the people you’re most likely to share culture and genes with. So minor differences in language, culture, and geography become important signs of essential distinction from your historical enemy.
