I think it just really depends. There is no fixed rule to how PhD programs are supposed to work. Sometimes your advisor will suggest projects he finds interesting and wants to see done, he just doesn't have time to do it himself. That's pretty common. Sometimes advisors don't have that and/or want students to come up with their own projects proposals, etc.

It depends on the program, and even more so, the student and the mentor. It can also vary over time, with more direction early on in a graduate program, and less direction later. Some mentors are very directive, and basically treat students as labor executing tasks they don't have time or want to do. Other times, the student is coming up with all the ideas and the mentor is facilitating it with resources or even nothing but uncertain advice or permissions now and then.

This can lead to a lot of problems as I think in some fields, by some academics, the default assumption is the former, when it's really the latter. This leads to a kind of overattribution of contribution by senior faculty, or conversely, an underappreciation of less senior individuals. The tendency for senior faculty be listed last on papers, and therefore, for the first and last authors to accumulate credit, is a good example of how twisted this logic has become.

It's one tiny example of enormous problems with credit in academics (but also maybe far afield from your question).

It is a spectrum. My advisor was very hands off. He didn't, ultimately, even really understand my PhD. He knew the problem, but he had no path in mind to solve it, that was up to me. I'm now working (as a software engineer) with a person who is very hands on with his students (and even postdocs) to the point of giving them specific tasks to do and then discussing the result every week. He defines the problems and structure of the solution, the students at least partially are an extension of himself, they are doing stuff he merely doesn't have time to do himself.

And there is everything in between.

Often at the start yes. So the students gets a bit of recognition, a bit of experience and a bit of knowledge.

From the cases I've observed directly in the area I work in, yes.