What could/do applicants usually ask professors, and how does this reflect in a statement of purpose?

1. 1. Can I ask more about what you research?

      Don’t be so vague. You can find out for yourself what a professor is researching by reading some of their most recent publications. Think about the papers carefully and try and ask a pertinent question, for example:

      I was reading your paper X and have a question about Theorem Y. Is it related to Lemma Z as described in paper Q?

   2. Do you think I am a good fit for the university?

      Instead of asking what the professor thinks, I would emphasise why I know I’m a good fit for that university. If you don’t like the look of the department, why bother contacting them in the first place?

   3. Would you consider taking me on as a PhD student?

      I prefer “Will you be taking on a PhD student next year?”, although this may be just a cultural/ politeness thing.

   4. Do you think I am better suited for MS/MPhil than a PhD?

      Again, I feel this is something you need to decide before contacting them. Look closely at the program requirements and think critically about your own skills and knowledge and where you will fit best. If you’re still in doubt, contact the admissions tutor with this question, not a potential supervisor.

2. Many of the PhDs I’ve been applying for lately strongly encourage inclusion of a potential supervisor's name in the statement of purpose, so they can direct the application to the relevant person. You could include a mention of the professor you contacted in the following way:

   My research interests lie in the field of pure maths, and more specifically how Lemma Z can be applied to Theorem Y. I would like to work with Professor A in research group B.

All of this is based on my own experience of applying to UK physics PhDs (which I am still in the process of doing).

I think you have misunderstood the advice you were given.

When you are advised to make sure your research interests align with those of some of the people working at the place you are applying to, people typically expect you to do the work yourself, not delegate it to random faculty members. If I received an email asking me what my research interests are, I'd either bin your email or, if I'm feeling generous, redirect you to my easy to find personal webpage where I've listed my research interests in great detail.

What you should do is go to the webpage of the institute/school/centre/unit/whatever you are applying to, and look through their staff directory. For each faculty member, there will often be a short summary of their research interests, and/or a link to their personal page, and/or recent/important publications (publications can be found in many other places as well, e.g. Google Scholar). If you cannot figure out the research interests of a faculty member using any of this information, perhaps graduate studies are not for you.

Or if that's too tedious for you, you can always look at research groups first to narrow down the list of faculty members. In this case, go to the webpage of each research group affiliated with the place you are applying to, and there should be a summary of the group's interests/goals somewhere there. If that information is not enough for your needs, but so far you think that group may be a match, go through the staff directory of the group and check the research interests of each group member. And if you want to know the specific things the group is working on, check the group publications.

So, contacting faculty for research proposals is inevitable, but completely unnecessary for research statements.