We have flown a few times with Air France and KLM.
The covid requirements mean you can check in online but you won’t be issued a boarding card. You will be allocated a seat however.
You need to pick up your boarding card at the check in desk at which time your forms / vaccination status will be checked.
It does mean a queue at check in and can be quite slow if you have people in front of you who don’t have all the right paperwork to hand.
You can show documents electronically or on paper. Bear in mind you might not have wifi access so don’t rely on showing documents held online make sure they are on your phone.
The documents you need depend on the country you are flying from and to.
For return to the UK your passenger locator form is linked to your passport if you are a UK passport holder so is automatically checked at border control when you land. Vaccination status isn’t checked on arrival just by airline when you get boarding pass.