Will I encounter issues with airlines or border control traveling to Thailand and back after Vietnam on a visa exemption?

You need to check your visa requirements for Vietnam..

Not sure on French passport but many passport countries must have a visa before travel.

You get 60 days on entry and are leaving before that date so not going to have a problem.

With a French passport (you and your girlfriend) will get a 60 day visa exemption when you arrive in bangkok. Because you have booked a flight out of thailand within the 60 days to Vietnam then you should have no problems with your airline or in entering thailand. If the airline ask just show them your booking to Vietnam this should satisfy them. If you wish to come back to Thailand after your visit to Vietnam then you will get stamped in again on a visa exemption for 60 days ( your flight from Vietnam may want to see a flight or travel onwards from thailand within the 60 day visa exemption in the same way you have done in entering into thailand the first time