SW side falafel; Pulaski Rd. pizza
The area around 63rd and Damen used to have a sizable Arab community. I believe there still may be some places there (Rene?).
Further south and west, in Bridgeview, is another sizable Arab community. There are several resturants along Harlem at around 110th. We tried one, once. It was good, but for us not really worth the trek (even if Krispy Kreme was on the way home). Since I have a limited knowledge of the Bridgeview places, I'd love to hear more specifics.
re: Vital Information
the following web page might have some leads for you. It lists a few places further west on 63rd. As well as a few in bridgeport. I have not tried any of those.
A long long time ago I remeber a very good syrian bakery on 63rd and about western. Our family would generally stop by for lavaza - the very large, very thin flatbread
There are definitely a number of small restaurants on 63rd Street, I think almost all west of Western (2400 W), where youd be likely to get some decent falafel. I think I ate at Steves (in Zims list) several years ago but there are some others not in the list. I think the highest concentration is around Kedzie (3200 W). Im less sure about 59th Street and unfortunately I know almost nothing about Bridgeview (its outside my usual foraging area). Maybe the bakery Zim mentioned is Al-Ahman at 2513 W 63rd?
For pizza, peppernegg, etc on Pulaski two places come to mind. Vito & Nicks (8435 S Pulaski) is one of my favorite places for classic Chicago-style thin crust. They only serve whole pies, not slices, but they have a full menu that might include eggplant parmigiana, and peppernegg (havent had either). Dont ignore the egg pizza (maybe with extra green peppers?) thats served on Friday only. Tonys (7007 S Pulaski) is a very good little beef stand where you can get your Friday peppernegg.