That's a lot more certain than the hope a neighborhood will improve, and usually a much higher rate of "appreciation".

My understanding is that you take the "improvement" (meaning - exclude the land portion) part of your property and depreciate over 27.5 years.  

More so if the house is older as you suggested and you can do some work to improve it.

Has there been extensive physical improvements such as a total rehabilitation of the structure?

Paying cash for the 1st rental is key and it is great you have the equity from your primary to do this.So I spoke with one lender that deals with investment financing and this is the method.  1: buy house with cash2: remodel and improve with intention of buying into equity3: (this is the tricky part with time)  Wait 6 months (some make u wait 12mo)  and get reappraised with intention of financing for 70% the appraised equity.4: Use the 70% pulled from 1st home to buy your 2nd property.This method works better in flips because once the 1st home is sold you can do the method over agian.  

Besides their proximity to an improving area I'm not sure if they're good places to look or not.What are your favorite (or not) Chicago neighborhoods for investing right now?

Looking at Multi-Family homes where to start he will live in on unit an landlord the rest while we improve the property, then move on.Option 2.

Finding a building where you can raise the value with general improvements or with better management is going to be the best suggestion I can give, but making the first step and getting "into the game" is the most important step.

Insulation and ventilation is the other answer,  however these solutions can be very difficult and expensive.  

We did quite a few improvements.