When I launch Red faction from the launcher/setup page. A window appears, that says, that the game needs 8 MB memory. When I press OK it Closes the game.
No problem with Red faction II.

The PCGamingWiki page says that this issue is due to the game's native incompatibility with modern Windows, so I'm surprised you are getting this as this is what GOG fix for these releases.

http://pcgamingwiki.com/wiki/Red_Faction#At_Least_8_MB_of_Video_Memory_is_Required

Still, you can try the fix suggestion on the PCGamingWiki page, that if installing the Pure Faction patch. Hope it works for you.

http://www.purefaction.org/

Thank you.

Installing the patch and launching the game from the patch, no problem.

Then when I tried to launch it from the original link on my desktop. The launch menu starts and when I hit Play, same old problem.

I found out, that if I bypass the launch menu (RedFaction.exe) and start the game direct (RF.exe) I have no problem. So it is the lanch menu, that is the problem.

That probably would have worked from the start.