Bochi

your french passport give you rights to stay for 90 days in the us as a tourist, if the us immigration received your K1, you gonna have to stay until a fiance visas is available for you, and with a fiance visas you have 90 days to get marry in the us to stay as a legal us resident. my advise for you is, if you decide to live in the us, better get married before you come to the us, and you'll be a us citizen husband, and with your 90 days visas tourist your french passport give you, you can before 90 days filling I-130 (Petition for Alien Relative) if you approved, you'll be a legal resident in the us.

I-130, Petition for Alien Relative, as a visa number becomes available. Filing and approval of an I-130 is only the first step in helping a relative immigrate to the United States. Eligible family members must wait until there is a visa number available before they can apply for an immigrant visa or adjustment of status to a lawful permanent resident.

If you live with your family and filing together , you need to file G-325A, (Biographic Information) for you not the kids, maybe your wife, so i encourage you to file that for your wife also anyway its free.