On Wednesday, July 3, 2013 1:32:22 AM UTC-4, nestork wrote:
1. In ANY fridge, the way to check the condition of the fridge is to
look at the frost formation on the evaporator coils. If this is a
frost free fridge; that is, one with a separate freezer compartment
with it's own door, then there will normally be a removable panel in
the freezer compartment, and the evaporator coil will be behind that
removable panel. If the fridge is in good health, then you should
see frost forming uniformly over the entire surface of the
evaporator coil. Frost forming only at one end of the evaporator
coil indicates a low refrigerant level (suggesting the possibility
of a leak), or a worn out compressor, or both. Sometimes, the
refrigerant charge will be so low that a "snowball" forms at the
inlet end of the evaporator coil, and the rest of the coil is frost
free. Seeing a uniform development of frost over the entire surface
area of the evaporator coil is proof that the fridge is in good
mechanical condition.

I pulled out the back of the freezer and photographed the evaporator fins. While the frosting is anything but uniform, it doesn't seem to be biased toward the top/bottom or the left/right.

Evaporator front view:
http://i42.tinypic.com/aov0x2.jpg

Evaporator rightward view from left side:
http://i42.tinypic.com/dmw6mb.jpg

Evaporator leftward view from right side:
http://i42.tinypic.com/jkhy8o.jpg

On Wednesday, July 3, 2013 8:26:59 PM UTC-4, nestork wrote:
The refrigerant in a fridge's cooling system will carry an oil mist
with it. If you ever puncture any of the refrigeration lines in an
old fridge, you'll get a lot of refrigerant coming out, but you'll
also see some oil come out with it.

So, the only thing I can imagine is that if there is a refrigerant
leak, some oil will also come out with the refrigerant, and that may
be what you're smelling. But, if that's the case, then the fridge
will stop working as the refrigerant escapes.

Maybe try and stuff some paper towels into areas you can't see to
see if the paper towels get oil on them. When you have a
refrigerant leak, all that appears to leak is oil because the
refrigerant evaporates into a vapour, leaving only the oil mist
inside it behind.

The only other thing I can think of is that if it wasn't a new
fridge, it could be that there was dust on the condenser coil or
condenser fan blades, and that perhaps that dist contained allergens
that you are sensitive to, causing you to cough. If you have good
access to the condenser coil and condenser fan, you might want to
try unplugging the fridge and cleaning the coil and the fan blades.

I also shoved paper towel around the compressor & fan. No signs of oil. On the fan, I tried to focus on the shaft/bearing area. On the Compressor, I tried to focus on the seam going around the compressor, as well as the inlets/outlets (whatever they are). Just for kicks, here are the photos before I wiped/vacuumed the area.

Compressor, fan, condensor, Front view:
http://i39.tinypic.com/kapizm.jpg

Compressor:
http://i43.tinypic.com/2dmf2p5.jpg

Condensor:
http://i39.tinypic.com/2uz2r28.jpg

Fan:
http://i43.tinypic.com/wsk7td.jpg

I guess I've done what I can. Thanks for your advice and for sharing your experience.