Can someone in the IRR run/get elected as President or any other position in the cabinet?

There is nothing under the "terms and conditions of employment" (applicable section of MILPERS blah blah) that outright precludes it. If you were AD or reservist, there's a thing where you have to avoid conflict of interest and obtaining permission if you're AD to hold an outside job. You'd have to punch out to run for elected office because you'll make yourself unavailable to perform your regular duties. A number of legislators are in the Reserves so they're able to juggle it. Re: Cabinet positions; that's a different story. If non-DoD, there's no conflict of interest. These positions are appointed. I'd expect the pedigree of folk in the Cabinet to have several College rags and considered preeminent in their line of work. We see that in various appointment of retired flag officers. Since the appointment is to a civilian position, then an AD would have to punch out while a reservist or IRR would be eligible. I just don't recall seeing that.

They can, and until elected or confirmed in office, they can remain in service. However, I believe that once that status changed, then they can defer or become discharged from their duty.

What law or regulation prevents it? IMO, they could, and if elected have to resign or be released from the military.