1. EMU can be a vague term, but often it refers to dedicated beds on a neurology floor for long-term continuous EEG monitoring of epileptic (or PNES) events for medication titration, seizure localization, and/or pre-operative planning. There are no patients on a ventilator in such a unit. In an academic center, residents and/or NPs cover the EMU and the epilepsy team makes decisions on medication weaning and EEG interpretation.
2. Continuous EEG monitoring for status epilepticus typically takes place in an ICU, hopefully a neuroICU if the hospital has one. I guess if only a few ICU beds were wired for continuous EEG, then you could call it an epilepsy unit, but it's fundamentally just an ICU. Sticking an IABP in a patient in the MICU doesn't turn that room into a CCU bed. The idea that an epileptologist would endeavor to manage a ventilator makes me cringe, and no hospital would/could ever credential an epileptologist without concomitant critical care licensure to manage ventilated patients as a primary service. Neurointensivists handle SE all the time, and in my center the epilepsy service is never formally consulted in the ICU. They do all the EEG/LTM reads, and we chat with them about longitudinal changes.
3. The job you take after fellowship could be any of the above, depending on your location and job preference. An epileptologist, at least in theory, has the skill set of a general neurologist underneath their specialty training. As such, they should be able to manage general neurology problems and even stroke. Most hospitals in this country do not have a dedicated stroke service staffed by vascular neurologists. You can seek out opportunities that have more or less inpatient responsibilities, but outside of big centers and without a big research component, you would probably have a hard time seeing only seizure patients in clinic, particularly early in your career.