After the creation of the bottle and handing off the installation of the game to Steam, Steam started up. So did Steamwebhelper except that it crashed almost immediately, was restarted by Steam to then crash again ad infinitum.
Changing the bottle from Win 10 to Win 8.1 fixed that and Conan downloaded happily. Starting the game up, however, started several copies of the downloader until I realised that I had to switch back to the Win 10 bottle.
I've run around a bit with my existing char and it seems to work ok and I created a char that worked perfectly fine as well. I will, however, need to play a bit more to see whether it runs as well as shown in the ratings.
One point for people who dislike the Steam DRM as much as I do and would prefer not to run the game through Steam: once it is installed, Conan doesn't need Steam anymore. I ran the .exe directly several times and it started up without Steam being involved.