Assuming I did this correctly heres what I got:
(Mutimeter set to 200 Ohms?)
You said to test in the On III position, but the diagram said On was II and Start was III so I did both

On II
(3)-5= 01.5
(3)-(1)= 00.5
5-(1) = 00.6

StartIII
(3)-5= 00.5
other two same.

 

Reconnect 5P and and 7P and try to start the car. Does it now start?

If not, use your multimeter to measure voltage across the two battery posts. Repeat test by measuring across the two battery connectors. Post the two voltages.

 

Started right up no issues. Its very random when it does it.

Checked battery voltage anyway
12.2 no matter how/where I probed it.

 

Next time it won't start, redo the three continuity tests.

 

Will do. Ill post back when it happens.

 

Update!!

After driving the car all week, and it behaving perfectly for around town runs, today I go to hop in and go pick up some miscellaneous parts for some maintenance on it, lo and behold doesn't start and everything is dead. Busted out the multimeter and dove right into the testing:

First thing I found was that when I unplugged the connector from the fuse panel (5P) the door chime came back on when the key was left in the ON-II position, shut back off when I plugged said connector back into the panel. When the chime was on the dash was still dark, no radio, etc etc.
Door chime went away when I unplugged both the 7P connector and the 5P connector.

ON-II
(3)-5: 0.05 + Continuity Tone
(3)-(1): Nothing
5-(1): Nothing

Start-III
(3)-5: 0.05 + Continuity tone
(3)-(1): 0.05 + Continuity tone
5-(1): 0.05 + Continuity tone

After poking everything I plugged the two connectors back in and tried to start it, no change.

Edit:
After letting it sit for a few minutes, went back out and tried to start it again, nothing.
Popped the hood and gave both battery terminals a couple good smacks with the handle of a screwdriver, turned the key, started up no issues.
Very confused now...