Pepperoncinis? You mean giardiniera? Or sport peppers? (The former for the beef, the latter for the hot dogs.) I think somebody might look at you funny if you asked for pepperoncinis at a hot dog joint.
I may be snobby about this, but I wouldn’t just go to any random hot dog stand. There’s a lot of mediocre stands, but I tend to be picky about my beef and hot dog spots. Beefy’s on Harlem is not far from Midway, and they serve up a respectable Italian beef, as well as a proper natural casing Vienna Beef hot dog (not the skinless crap they pass for hot dogs at many joints.) Their version of the full “Chicago style” is a little different, as it also includes fresh cucumber slices on it (which is not part of what has become the canonical “Chicago 7” toppings. That said, I’m not partial to the fully dressed dog, myself, having grown up with a pared-down Chicago dog.) There’s also a Portillo’s there on Harlem, which local food afficianados seem to have a love-it or hate-it relationship with, for some reason (probably because it’s become a national chain). That particular location is good and will give you a fair idea of a Chicago hot dog. Now if you’re flying out of Midway, Gold Coast Dogs in the airport serves a respectable Chicago dog, as well.
The perfect place for you guys for an Italian beef with an 11-year-old in tow would be Al’s #1 on Taylor Street, as it also has Mario’s Italian lemonade across the street, which the kid will love. The problem is you would probably need to Uber or Lyft or cab it there from downtown, as it’s not the easiest to get to with a compressed schedule.
Downtown, the best place for Italian Beef is Luke’s, 215 W. Jackson.
If you’re staying in that hotel complex on Cicero, Lou Malnati’s is a solid representation of Chicago deep dish. They’re pretty new to the area there, but have a long history in Chicago as one of the first. If you want to try deep dish, that’s solid. There’s also Giordano’s in the area, but they’re stuffed pizza, a later (1970s) offshoot of the Uno’s/Malnati’s style deep dish (which goes back to the 40s or possibly earlier.)
Oh, and I second @bobot 's suggestion of the architectural boat tour if you can swing it. If it’s a nice summer day, it’s a great way to get a perspective on Chicago you normally wouldn’t, along with some interesting architectural history. I went on it for the first time when I was 22 when I was dating a German woman, as I thought it might be a fun, touristy thing to do, and found it quite informative and entertaining myself.