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More photos.  The class might be helpful.  

Taking a look at the wiring behind the light assembly, makes me think something is going on there....not sure what yet.

Well using the British made tool, it says I am a mark III.  But i have the quarter windows and innocenti bits everywhere (Guages are italian, strang turn stalk/headlight).  

Yeah the engine is a 99HE20 which according to this thread at NorthAmerican Motoring means I have an A+ engine from 1985 onward.  Explains the transmission swap!

Here is the guides posted here on identifying your model year

//www.minimania.com/Mini_Identification_1837

//www.minimania.com/Mini_Identification_1807

if you need a wiring diagram then let me know what year and model you have

Don't bother with the engine...those are swapped out too often.  I would look at the codes on your door locks, trunk lock, and wondshield wiper motor.  If I remember right there will be 4 numbers...a two digit week code and a two digit year code.  I think that the lenses you posted pics of are for the 1977+ models with the integrated reverse light.  If your car is prior to that date then things could be wired badly from when these lights were installed by a po.

I believe it is titled a 73

The paint code is Rosso 72

Let me go grab the engine block numbers.  What else should i check.  Given that the engine USED to be remote shift, and Innocentis in the +/- incorporated MKIII/IV depending on the parts bin

In short, yes.  I have run into problems with the lower fuse in the 4fuse box being loose and not having working tail lights, but I think your problem is likely different.  You might first verify that those lenses are correct to your car.  Some older cars retrofitted newer style lenses because they integrated reverse lights.  Have you done a check to see what year your car really is?

This might be a dumb question, but should my car have rear running lights at night?  I have side marker lights up front (yellow), but nothing in back except for reverse and brake lights.

My lights look like this

//www.minimania.com/part/XFB101200/Right-Rear-Tail-Lamp-Light-Assy-Including-Reverse-Lens

Top to bottom?

Yellow: Blinker

Dark Red: Brake Lights

White: Reverse

Light Red: is that even a lamp or just a reflector (like a bike???)