How does Firebase work with PHP?

1. No you don't need MySQL if you are using Firebase. But, there are good reasons to use one or the other, or both together. They are very different in how they work and you need to evaluate them independently for your use cases.
2. Not necessarily, because Firebase is "schema-less", meaning it is based entirely on the JSON data you put into it. In essence, it's a massive JSON object you put data into. (But, I'm not saying you don't need a little planning to make things work right, I'm merely saying you don't need to plan all your data types, columns, etc.)
3. Maybe, maybe not. It depends on how familiar you are with JSON, working with schema-less data, and working with restful services. I would argue that it all depends on what you're trying to accomplish, and how complex your system is, as to whether it will speed things up or not.
4. No (but maybe yes). Are you building a Node application? If you're building in Node then you can do everything through Firebase (they can host it for you). If you are delivering the first byte in something else (like PHP), then you would need a different host like AWS or Rackspace. But, you'd always have Firebase hosting your data.

Hope those answers help! Please comment under this answer if you need any clarification.