Cooking Fumes, Cleaning Products, Bird Feeders & Parrotlets

I have only a small apartment. My main living area is a combined
living room / kitchen that is about 20' x 15'. If I keep the
parrotlet's cage on the opposite side of the room as the stove (as far
away as possible), are cooking fumes still a risk? I typically run
apartment fans and leave the windows open while I cook.

Cooking fumes themselves are not generally harmful - overheated nonstick pans are as well as smoke. I am not in your home and don't know how it is when you cook but if the area is well-ventilated and you're not overheating pans or making things that generate smoke, I would imagine it would be ok. Many people keep parrotlets in apartments and manage to keep them safe and healthy; you will have to monitor the situation and make adjustments as necessary.

I anticipate cleaning my parrotlet's cage with Clorox "Green Works
Natural Glass and Surface Cleaner." Once a week, I plan to remove my
parrotlet, spray the cage with the cleaner, wipe off any obvious stains,
and then give the whole cage a shower. Would this cleaning procedure be
safe for the parrotlet?

I do not use any chemicals on my birds' cages. I am old school and good old soap and water that is well rinsed is perfectly fine for most cleaning procedures. Chemicals not only produce dangerous fumes but if not completely rinsed can kill the bird. Soap and water is just as effective for cleaning and doesn't involve the risks of chemicals.

I clean my apartment once a week, and I use the same Clorox cleaner
mentioned above to clean all my surfaces. I use a harsher cleaner for
my shower, but I keep the bathroom door closed and fan on while doing
that cleaning. If the apartment windows are open and fans are on while
I clean, will the parrotlet be safe from fumes?

You should check with the manufacturer. I would imagine these chemicals would be very dangerous for birds but I don't use them so I don't know.

I maintain a few bird feeders outdoors, and I bring them in to clean
them each week. If I wash my hands several times after cleaning the
feeders, should I be concerned about transferring germs from the feeders
to my parrotlet?

I would think if you washed everything thoroughly and changed your clothes and shoes after cleaning the bird feeders, you would be safe in preventing disease transmission.