Every adult neurology residency I interviewed / looked at had a dedicated inpatient service. I can really only describe my program in depth, but the gist is similar at most programs based on conversations with friends elsewhere. I'm sure there is a range, but at my program we deal with what I would consider a lot of medical problems. We consult medicine specialty services (e.g. cardiology, nephrology) when needed, but we manage simple AKI, Afib with RVR, post-stroke aspiration pneumonia without medicine involvement a majority of the time. Some programs probably handle less medicine and have an easier time transferring to a medicine service, and some may handle more.
MICU responds to all code blues, but obviously won't be at bedside right away, so you're managing very sick patients on the floor / step down until you can get them to the NSICU (which is semi-closed at our institution, so once they are there, a separate NSICU team handles direct patient care).