Q1: What's the exact difference if I either download the real GSX installer (not for the first time, of course) or the standalone Addon-Manager?
That's very simple: the Stand-Alone Addon Manager doesn't contain GSX! As indicated on all Info pages of all products, including GSX:
If you want to update the Addon Manager to the latest version, without having to download the whole product again, you can use the Addon Manager Standalone installer:
The Stand-Alone Addon Manager contains lots of support files for both sceneries and GSX, but doesn't contain what makes the bulk of GSX itself, which are the vehicles with their animations and their 100+ liveries and all sounds. That explains the size difference. Of course, just installing the Addon Manager without GSX will not result in a complete working GSX. But not the opposite: installing GSX with the full installer DOES contain everything the Stand-Alone Addon Manager contains so, it doesn't require anything extra.
The only need to install the Stand-Alone Addon Manager AFTER having installed the GSX full installer is:
1) There's a major update (GSX will tell you)
OR
2) You installed a scenery using an old installer after GSX, so you want to be sure you have all support files updated
OR
3) You have a problem in your Windows install, like the MS VC++ 2005 runtimes missing/corrupted that result in GSX not working, and since the Addon Manager that comes in the Stand-Alone installer requires a *slightly* different version of those runtimes (there are many versions around and they can all be installed alongside each other) that it's hopefully not missing/corrupted on your system so, chances are that one will run better.
Q2: Why is the Addon-Manager still about 14 MB in size, though it downloads its required data for installing from the web anyway?
So we can update some important files without telling ALL users "please redownload the Stand-Alone Addon Manager again", because it would be enough they'll run the installer they already have. Out of 14MB, about 4-5 are downloaded from the internet. And, there's a copy of them IN the installer too so, in case you don't have an internet connection, you might not get the very latest files, but you will still install a complete Addon Manager.
Q3: May an use of the wrong installer possibly open opportunities for errors to appearing?
There's no such thing as a "wrong" installer. All installers on our site are always the most current versions and, since they download some files from the internet, they have some self-update capability.
Q4: Does the live-updater do the same and vice versa?
That's of course explained in the GSX manual. The live update it's a GSX feature, and there are two kind of updates: minor updates that can be applied only the fly without restarting FSX, and major updates that requires to exit from FSX, download the Stand-Alone Addon Manager and install that one. GSX will clearly tell which kind of update you'll eventually need.
