How did they make the cars fall apart, on cue, in those old comedy shorts from Hal Roach Studios? example: Blotto with Laurel and Hardy
Many of the special effects created for the Hal Roach era were created and executed by Roy Seawright, considered one of the greatest FX artists of all time.
He had extensive skills in all aspects of special effects, including matte paintings, optical effects and on-set practical effects.
Seawright was responsible for some of the most memorable car gags, including the splitting car effects that you mention in your original question. These were practical gags, performed using specially constructed cars that were designed to break apart on cue.
These amazing props, combined with clever cinematography (consideration of angles and lighting) resulted in the classic scenes that still delight audiences today.
A little more information about Roy Seawright can be found here.
Here is one of Seawright's prop cars - a crushed model that still drove.