To show interest is never wrong so from that point, I think such an approach would be fair. What complicates the issue is the picture of prerequisites. When you approach someone, does not matter at what level, you need to show how you can be of interest. What that entails is showing you have the background (courses and scientific literature) that allows you to be efficient help. I think this is where the plan is most likely to fail because the recipient of the request will not likely be interested in taking on help that needs much coaching to function. If your request is within the department where you study, matters might be easier but if you contact someone in another university of perhaps even department the difficulties may arise.
So, I cannot see anyone being offended by a request such as this. The problem lies in seeing how you can fit in and be a contributor without to much costs (in terms of e.g. time) for the researcher or research group. However, a personal visit is far more likely to lead somewhere than a letter or E-mail.
Requests such as these are not uncommon but in my experience have most often been very uninteresting. Again, some have visited which has provided a very good possibility to assess the common interests and in what way a possible "collaboration" would work.
As for the last question, I cannot see a problem except that a PhD student may not be in a position to bring in an external person into a project that in essence is run by their advisor. A PhD student, on the other hand, may be a good way into a research group since they may be open to a meeting more readily than a busy project leader.
So, showing interest is good but you need to be able to show clearly what you bring to the table to wet the appetite of the person you target.