The fuses are notorious for causing all sorts of problems like this. They say you should just change all the fuses. I am in the midst of doing just that because of my car being rendered inoperable by one of them going bad. They may look ok but they can crack and cause sporadic problems. You may also want to clean the contacts with some fine sandpaper and use dielectric grease to prevent oxidation. Fuse #5 on W124 is the instrument cluster among other things and the alarm is wired into it possibly causing the sporadic non-starts...
1993 300e-2.8 - gone now <sigh>
"Do not adjust your mind, it's reality that's malfunctioning"