I'm hopping someone can offer a little, or a LOT, of help.  My '82 CE w/crossfire, intermittently will not start.  When I turn the key, the dash lights, gauges, radio all come on, the horn will honk, etc.. but it will not crank.  It is like it was left in gear.  The starter and solenoid have  been rebuilt, the wiring harness to the starter has been replaced, the ignition switch has been replaced.  And the theft deterrent system has been removed.  (it was never connected when I purchased 4 yrs ago).  I drove to work around 8:00AM today and parked on the street, with a slight downhill grade.  At 10:30AM,  I went to move into the parking lot and it would not start.  I moved the shifter, rocked the car and let it roll forward.  Still no crank.  At 12:30, I tried and it started and ran great,  (straight to the shop).  It had sat for over 2 hours, (out in the sun), and wouldn't start.  2 hours later, and hotter, it started.   Any and all help/direction is greatly appreciated!!  I just want to drive!!

Sounds like you've pretty well covered most of the basics on this, with a couple of exceptions...hopefully one of these will be the answer...

Have you tried starting it in neutral, or in any of the other gears? You may have a bad/failing neutral safety switch, located on the shifter, under the console.

If the NSS is ok, then the last thing I can think of is the firewall/fusebox pass-thru. The terminals inside can get pretty corroded over the years, and create intermittent starting issues, as well as other electrical bug-a-boos. You may need to dis-connect the connector there, and give a really good look. You may even need to clean some of the terminals, if you see any chalky green/white or rusty stuff in there.

I have to assume that you have already checked the battery cables, and grounds?

I have tried to start in all gears, nothing happens.   The shop said that the neutral switch checked out ok.  Battery connectors are clean and tight.  We'll check the firewall/fusebox pass thru.  I can't help but think it is heat related, except, it is intermittent.  The battery is 5 years old, (72 mos battery).  Do you think a  defective battery could exhibit these symptoms? Just not sure where to look next.

The connector that Joel noted caused mine to cut out and then not start, no power to anything. It was corroded, took it out & hard wired it.

I don't know if this is your problem or not.

Also, check the 20 amp fuse in the battery compartment - a red wire from your positive cable will run to a terminal block and this fuse is attached to that. This fuse, fuse plug and terminal block can get very corroded over the years from battery acid fumes and cause exactly what you are saying. 
  Try cleaning up this fuse housing and replace with a fresh 20 amp fuse. Also clean and check all the wires and connectors that connect to that block terminal. I replaced the block on mine and used dielectric grease on the fuse, fuse plug and all connectors to the block.  The power from this block/fuse runs to the ECM computer and if it is not happy, it will not allow the car to start.
  If all this and these other suggestions don't work, it could be the ECM is going bad, but most likely not.
Good Luck!
 

This could also be a bad starter solonoid.  If you have power to the purple wire to the solonoid and it won't start, replace the sol.

Unfortunately, the best time to check it is naturally when it won't start, take a test light and put the lead to a good ground, on the starter check to see if the big cable going to the starter has power to it, next have someone turn the ignition to the crank position and see if the is power to the small terminal...the "S" terminal. And all of the above.