Hi, not aware of any customs in Cornavin station. We crossed into France a few times without any border issues.

Check out the SBB website - they have a detailed list of stations and facilities.

There is a facility, not sure of hours. Check the excellent SBB website which lists facilities at each station in Switzerland.

If arriving at Geneve station there will not ne a problem , look at Seat 61 Com ( tripadvsior will not allow links) there is plenty of info for you on there .

I am not sure if the station stays open overnight during the week at weekend trains run through the night so it will be open. That said, I wouldn’t recommend staying there. It’s a city centre station and with it goes the usual issues of drunks and pickpockets, especially at night. There are plenty of reasonably priced hotels next to the station such as the IBIS. Hope this helps

Unless you want to catch a flight

10 min and do not forget to take a free ticket for travel near baggage conveyor I think it is 10 .