1. Will a modern Monarch hold Court?
No, as this is something handled by the judicial system (courts) these days. It doesn't mean they can't in game, however.
2. Is there any occasion at all except coronation that the Monarch will actually wear the crown?
When sitting for official photoshoots or portraits, maybe. Maybe they like to be extra formal for weddings or press conferences.
3. If your Monarch hold Court (events), how do you play them out?
If you have your monarch in the Law career as a Judge then they could judge cases and make decisions on law breakers (I think this is done via the computer, so more of an admin/paperwork thing). This is a WFH career, so doesn't take much time daily and frees the sim up the rest of the time for royal duties.
If you have them be famous with a good reputation, they could hold fundraiser events for charities, read fan/hate mail (petitions?) and donate items to fans (subjects)/gift money to sims to help out sims that appeal to the royal house for help. You could also do meet and greet events at public venues, maybe playing it out as "opening" new buildings etc.
