Shengen visas I believe are just that - any shengen country to another - the original visa remains valid. IF you have a shengen visa. If not, you are down to the entry rules for that country depending on your nationality so its about checking each country individually. I am unaware of any differentiation between arriving GA or scheduled carrier anywhere other than the USA though.
>When a Brit flies to the USA, via GA, he does need to get a Visa.
Yes, and I believe there is a once only process/concession where they will issue a B visa if you have screwed up and arrive without one, albeit at vast cost. The only way to enter without a visa otherwise is if you have already entered on a visa waiver via a carrier, go out and come back in GA within the original validity of the waiver. Doubtless terms and conditions apply but it is on the border website that one can do that.