Context: I'm an older PhD student (mid-50s) and had a couple of publications before entering the PhD program. I'm in an interdisciplinary field where there are few well-established research programs. It's very common for PhD students to embark on their own research interests that are different from their advisers. Myself, I started submitting papers after my first semester based on class projects and side projects. I even developed a paper that started as my answer to a question on our Candidacy Exam. I was a research assistant for three years with sole responsibility for a very complex Agent-based Modeling simulation.
To what extent should a graduate student be dependent on faculty members? How can he be more independent and start his own research activities?
Start as soon as possible to develop your own intellectual curiosity. This involves thinking hard about the research in your field relative to the fundamental questions in the field. This is more than "picking sides" in intellectual debates. It involves developing mental models of your field, the various research methods, research questions, and research results. What's important? What is not?
You might start with a single important paper. Look at the section of the paper where they describe opportunities for further research, or maybe limitations. Read other papers that critique this important paper. All the time, you should be asking yourself: "What do I think should be done to improve/extend/solidify this research?"
You could also start with a survey paper, which often include extensive discussions of future research directions.
As soon as possible, you should put your intellectual curiosity into action. I'm fond of class projects and papers for that but not all disciplines have courses that support project work. Either way, "directed reading" courses are ideal settings to write conference or journal papers of your choice under the supervision of a professor (not necessarily your adviser).
Is it a good idea [to do] the graduate work on another topic with the faculty member or work independently on a more recent research topic?
If your intellectual curiosity leads you to research questions/methods that can be successfully done in collaboration with a faculty member, then approach them and suggest a collaboration. If, like me, your interests and ideas lead elsewhere, then do your research independently.
If the person is in a research group or at the university, is it a good idea or is it ethical that he develop his own research interests, besides to the topics which are being developed in that institute?
It's a good idea to develop and invest in your own research interests if they diverge from your research group. There is no problem with ethics if you are open and forthcoming about what you are doing and why. This might go against the cultural norms of the university or research group, but you should make decisions you feel good about in terms of your ethics and values. If other people aren't happy with them or push back, then my view is that is their problem. If they have power, you may suffer negative consequences. So be it. Don't bow before people in power just because they are in power.
How should a graduate student manage this and make some progress in the research fields of his own interest? (In particular, sometimes the faculty members are not interested in the topics you are interested in and the person has to start his own research activity.)
Make a work plan. What is your output (results, papers, articles)? Where will you submit it? In what sequence? How much time will it take -- daily, weekly, monthly? If you can't make a plan that answers these questions, recruit a mentor. It could be anyone who you respect and who knows your field, maybe faculty at your university or maybe someone far distant.
In general, the more you can weave your independent research into your other activities and projects, the better off you will be. That's why I like class projects. But same goes for research assistant work, industry work, or other work. Look for as much synergy between all these activities as you can.
[...] if this research is part of his independent research, [...] what affiliation should be used in the publications? Should it be "Independent Scholar" as mentioned in this link or something else?
No, you shouldn't use "Independent Scholar" as affiliation as long as you have an official affiliation as a graduate student, post-doc, or other. You should always use your primary affiliation, even if it is Physics and your paper is in Art History. The exception is if you have multiple official affiliations (e.g. Visiting Researcher) and your research was done at or was supported by these other affiliated institutions.