Does anyone know what size tire and rim would have been used as a spare in a 67-69 dart? Did the mat and tire cover fit flush to the trunk floor or did it stick up some?

I can't remember the EXACT size but it's smaller then the 205-70 tires I usually run on mine, like a E-78 14 or something close, otherwise it doesn't fit flush into the spare tire well.
When you have the right size the cover and mat fit slush so the floor is level, there's a tie down and all that for it too if you don't have all the parts anymore. I always tried to find the biggest one that would still fit flush when I still carried a spare, though with a powerlock sure grip the actual height is very important if you have to replace a rear tire.

As I understand it the car came with FIVE identical wheel/tire combos.

There were 5 of the same size tires, '67 & '68 will be on the fender tag.
As far as the cover & mat fitting - most will be flat, the 70 series tires on the performance models won't be flat.
'67 /6 - STANDARD tire is a 6.50x13 (13x4.5" wheel was the smallest)
'69 GTS - STANDARD tire is E70-14 (14x5.5" wheel was the largest)