downloading I get a message "ERROR when creating patched vehicles_airport, restoring from backup now."
You mean when installing, not downloading.
Have you changed the default vehicles ? This message is not an error, it simply means that GSX installer wasn't able to patch the default vehicles file, probably because it's not the original version. This is not a problem, the only reason to patch that file is to get rid of the default pushback truck, so you might see two pushback trucks (not overlapped, because the GSX one starts from the side ). So this is not a problem, you can ignore this message.
After installation, there is no icon for GSX either.
That's not a problem either, there's no GSX icon, because it's not a program you launch yourself, and it's launched by FSX automatically.
When I open FSX, the security checks asking if I trust the publisher don't show up either.
That might not be a problem, if you already trusted us ( "Virtuali s.a.s." ) as a publisher. But it IS a problem, if you mistakenly UN-trusted us, even in the past, so nothing signed by us will ever run on your system.
To check this, go to the Control Panel, under "Internet Options", look for the "Content" tab, there should be a "Publishers" button, that will list the software publishers that you trusted, and another tab with untrusted publishers. If you see "Virtuali s.a.s", remove it from both tabs (especially if it's under untrusted)
I can't access gsx when I start a flight either. Is this just a problem with the demo or will it also occur in the full version?
Check the problem with trusted publishers, then check you have BOTH the "Addon Manager" menu AND the "Couatl powered products" menu under the FSX "Addons" menu.
Later, I uninstall the gsx demo. When its done uninstalling a message comes saying "The vehicles_ airport.gsx file is missing. Cannot restore from it.
That's not a problem either because, obviously, since the installer didn't patched the original file when installing, it hasn't created a needless backup file for it (no point creating a backup if nothing was patched), so it's now saying exactly that: there's no backup to restore from so, nothing has being done.