Hosting a database is no different than self-hosting any other service. This viewpoint hath what cloud wrought, this atrophying of the most basic operational skills, as if running these magic services are only achievable by the hyperscalers who said they are the only ones capable.
The answers to all of your questions are a hard: it depends. What are your engineering objectives? What are your business requirements? Uptime? Performance? Cost constraints and considerations? The cloud doesn't take away the need to answer these questions, it's just that self-hosting actually requires you to know what you are doing versus clicking a button and just hoping for the best.