Is it ethical to post (part of) my research question on the internet?
Posting questions is ethical, the issue is with appropriate credit and potential collaboration. I would say that  most answers will be directions to published work that covers your question, if the question is about problems already solved. In this case, you just have to reference the books or papers that you used. 

If it turns out to be an open question, it probably won't be answered in a simple post on internet sites. In this case, in the best scenario, you could find people that will collaborate with you in a more permanent way, till you both solve the issue and, eventually, publish a paper.