I am a French citizen and live in the US and I want to apply for and use a Carte Bleu from Societe Generale where I have an account. Will they send it to my address in the US and will I be able to use this credit card to make purchases from the US?
You could use it in the US, but there are fees for using your SocGen card outside the Eurozone. I find the SocGen card to be unreliable for uses other than with a merchant. Online my transactions are declined unpredictably, and contactless payments are similarly randomly accepted or declined.
As for whether they'll send you a card to the US... not sure. When we got out accounts last year, it was mandatory to pick up your "moyens de paiement" (e.g. Carte bleue and checkbook) at your local bank office. They've now changed policy and will mail cards to your residence (I got a replacement card in December this way -- but I live in Paris).
Further complicating matters is if your are a dual citizen of the US. The US requires non-US banks to collect and share data on American citizens' overseas accounts, and banks are hesitant to bother with the extra paperwork.