I don't know how things exactly work in the building department, but I would guess that the inspector could only cover the scope of work that is in your building permit. So if your permit only covers your kitchen work, I would think that the inspector could only look at the kitchen and not your stairs.
Be forewarned, when the day comes when you sell your home, your buyer will most likely have a home inspection done. The home inspector will probably notice the stairs are not up to code. However, they may be grandfathered in and not have to comply with current building codes if they were code compliant when originally built.
In regards to carbon monoxide detectors, they are required for a home appraisal, regardless if you are selling or re-financing. If the appraiser does not see them, he will not complete the appraisal report and will have to come back (sometimes at an additional charge) once they are installed.
I would not count on the inspector sticking to the work at hand. Inspectors are people. They may come with an opinion, a duty, an agenda or an attitude. I've sold more homes than I can count and have been involved in many that had to have a permit inspection. On perhaps a third of them, regardless of county or what was requested, the inspector checked on other things, usually safety related, such as water heater (strapping, bracing, venting, and piping), electrical (GFCI, open junction boxes, unprotected wiring, etc.), fencing and more.
If you get away with keeping the stairs as they are and your home is inspected by a buyer, code may not be the only issue. The inspector will likely warn the buyer of why the code is there and the buyer may then want the stairs brought up to code.
I see these things happen all the time.
