For a postdoctoral/Ph.D. position in Mathematics in Europe, can I make this somewhat irregular request? I am not sure whether this question is suitable for academia, if not it would be great to know the site where I should ask this. I am a finishing Ph.D. student in Mathematics at a decent school in the US, and my postdoctoral job applications so far have gone in vein, although I do have published paper in recognized journal. Also, I had to switch my advisor and subject in the middle of my Ph.D., so I think I would have been better off with more experience in the field I am working on or a related field. That would mean even if I get a postdoc, I would feel a little bit inferior to other postdocs or might seek a little more help from my mentor.

I noticed the postings by some European Universities for postdocs where they mention that they are looking for postdocs and Ph.D. students both. Although I will be applying for postdocs in these universities, I was thinking of having your opinion on the following, somewhat irregular backup plan:

Say University U in Europe has a postdoc position P for salary x for 2 years, and Ph.D. position D for salary y for 4 years, where x>y. If I can get P, great, but if not, can I also talk to and try to come to an agreement with the university (e-mail them with explanation etc.) for a visiting position where I will be learning some more stuffs and will be researching as well, but may be with less salary than P but the same (or little more) salary as/than D? In other words, I feel like I need to gain more knowledge in my area, so if I don't get P, I will be happy with almost the same salary y as D, but will try to bend the rule from 4 years to just 1 year? So it will be like a visiting position with less salary, of less time period, but I will like to be treated like a Ph.D. student (that means less independent research, more learning compared to a postdoc). Will that be a ridiculous thing to try? So, I will be somewhat like an advanced Ph.D. student but not expert like a postdoc and so I am willing to negotiate with the salary, I won't need another Ph.D. since I will have had one(if they give another, fantastic!) Do you know whether it might work or create complications enough so that the hiring committee will be offended? Should I try to send an e-mail explaining this situation and see whether they will negotiate with me on this?

Thank you very much in advance!
In several European countries, for example Germany, Switzerland, or Austria, it will be possible to fill positions in part-time. Whether you then actually work only part-time or not is a different question. In these countries, there is in most cases not a big difference between PhD students and post-docs regarding organizational details of employment, or even salary on a full-time position.

So the way I see your plan possibly working is as follows: A professor has funding to employ someone full-time for 2 years. You could then try to argue that in addition to working on the particular project, you also need time to educate yourself further and get only a half-time position, but stretch that to 4 years. You may want to communicate that it still means that you are working or studying full-time on project-related topics. If the funding is flexible regarding the year when it is spent (which is sometimes, but not always, the case), then I'd guess you have good chances that people are willing to make that deal.

This plan is probably best discussed with the professor offering the position, informally before you apply for the position, or when you go for an interview.