Sorry. Apparently I got confused by your mentioning of steam…
Is there a special way a .run file should be installed? I’m just clicking it or typing ./<filename>. Should I be root when this is done? The package manager requires my password to gain SU powers. Should I be doing this as a SU?
That depends on the installer of course. If it wants to install things systemwide, it would need root powers. But don’t see that mentioned anywhere on there technical support section, so I think running as user is fine and recommended.
Edit: No other output is shown in the console. Simply segmentation fault.
You could try to run it in gdb (you may have to install that first) and look at the backtrace to get an idea what could be wrong:
gdb *filename.run*
(gdb) run
...
(gdb) bt