[W126 Coupe] what's out of adjustment

[Shayegan, Richard]

I feel like there is something simple out of adjustment that is causing
these two symptoms, but don't have the slightest clue as to what:

Sometimes (probably like 1 out of 15 times, and there is no rhyme or
reason it seems), when I start the car, it will start up instantly (i.e.
the instant I touch the starter) and die just as fast. Then when I crank
it again it starts up normally

Also, sometimes, again, no rhyme or reason, the car will idle rough, and
goosing it a few times in neutral sorta clears its throat and smooths
things out. This could be exacerbated by the fact that I need motor
mounts.

Any ideas?
Richard