Cooking amenities are more important.

Not to mention the fact that if I want to cook it again, I will have to clean out the used charcoal.

If it were mine, I would do all of the following before shelling out money for a secondary system.Check ther attic insulation.Consider installing UV heat repelant tint (it has a service life) on all sun side windows.Have a thermo scan done of the exterior and interior to see where heat problems occur.Have the AC system check by a tech for proper amount of coolant, and to see if its too big or too small.Vent the cooking hood outdoors (if it's legal to do so).Install ceiling fans in the hallway upstairs and in the bedrooms.Have tech make sure the draw air system is sized (and positioned) correctly for each floor.Make sure the thermostat is not a (NASA Project of complication designed to micro manage the temp), and that it is working correctly.Some Techs are much better than others....Good Luck!

In general, the main issues that come up with city regs are adding a full kitchen (stove/cook=top), completely separating the units with a wall vs a door, and not having egress windows (big safety/liability hazard). 

Kilz oil-based or its equivalent paint is the way.

Does anyone have any recommendations for reasonably priced tax appeal services for Cook county?

If it's driven by a single industry or employer (ie shale oil in the dakotas) that's very risky and not sustainable.

It's been mentioned a time or two on here, but another option you don't list and that may be helpful for higher earning W-2 folks (or accredited investors) is to invest in oil and gas. 

@Dillon Cook, do you have suggestions on websites (or ones that you use) to find this information?

If not (ie two full-time working spouses) your only other option is going into land conservation deals, solar deals, or oil and gas deals - all of which have some risks.Explained in a different way...1) There are ordinary/W2/active income on one side.