tickets – Train from Geneva Airport to Fiesch

I see no reason why you could not do it. There are ATMs and workstations in the Geneva airport station. As long as you have money or an acceptable card, this should be pretty simple.

There are two trains per hour, so even if you have to queue a little, that should not be a problem.

However, I do not see any reason for not being able to buy tickets online. Have you tried sbb.ch?

Customs and Immigration – As a citizen of the EU who exercises his treaty rights, can I travel to France via Geneva with my family?

Let's just say: I am Danish and I exercise treaty rights in the UK, where I live with my Russian wife. My wife has a UK residence card because she's part of my family. Switzerland requires a visa (free and fast but boring).

Two questions, can we go to Geneva (the airline will she board)? Should we go out by the French part of the airport?