Is it considered ok to proactively ask a professor from a different university that you never met in person about some potential research assistant internship? Or is something like that considered impolite or is it a complete waste of time due to zero probability of getting it?
All of us get such emails all the time -- I get about 1-2 per week. Anecdotally, most professors just delete these answers, whereas some (myself included) at least cursorily look at the usually attached CV to realize that there is little overlap in research area between the person asking and myself, upon which I then write back and advise the student to apply to the graduate program of my department rather than myself.
So, just going by empirical evidence, it is ok for you to write such an email. But it's unlikely that you will get many responses and even more unlikely that you will actually get an RA position this way.
While @WolfgangBangerth's answer rings true, you can frequently "stand out from the crowd" of emails by paying careful attention to the subject line of the email and your first sentence.
- Research the professor's current work.
- Isolate what it is you find particularly interesting about their current work that genuinely captures your interest.
- Construct a subject line that honestly represents your question about potentially working for them, and simultaneously includes research terms you suspect will catch their eye. Perhaps of the form "Looking for research assistantship in the History of Molecular Psychology Tensors".
Then make sure your first sentence fits the same rules (honest and simultaneously academically catchy).
This demonstrates that you are not using a shotgun approach sending generic letters to dozens or hundreds of professor. You find them interesting, you find them special, and you'd really be interested in the possibility of working for them in particular.
Just make sure that you mean it and are being sincere.
Of course, you can email the professor. You can email anyone at any time for (almost) any reason. The worst that can reasonably happen if you send a single polite email is that you will be ignored.
I don't agree with Wolfgang Bangerth in the assessment that it is necessarily a waste of time. Not every professor is flooded with such emails; someone relatively early in their career might not be. But you will need to have something to offer.
A junior professor needs papers. Either from research they do themselves, or from research they supervise. If they get a good assistant who writes a paper with relatively little supervision, that's great. If they get someone who needs a lot of supervision but won't succeed in producing research and writing a paper, then they might not find it worth their time. Sure, supervising is experience too, but they probably have enough people closer-by if that's all they need. Papers count. Supervised n students resulting in 0 papers not so much.
Are you someone who can produce research and a paper with relatively little supervision? Can you convince the professor that this is the case? This is the tricky bit. Not many very early career scientists can do this. Writing a well-crafted (short!) email is good (see uhohs answer), but not enough. You'll need something to show. Have you done any really successful student projects, preferably related to their field of interest? Can someone more senior, preferably someone they already know, introduce you, perhaps write the email for you? Either of these will strongly increase your chances. Your email might be ignored. An email from their collaborator, I have this really great student who would love to work with you, would you have a moment to talk to them?, might not be. Of course that works only if you are great, otherwise asking for such an introduction may be awkward.
I was a research assistant during both Bachelor and Master. The Bachelor work helped prepare a paper where I became co-author, and the Master work in a paper I authored myself. Although I was at the same university in both cases, I didn't really know the professor at that point. At least for the Master, I don't think he was flooded with emails from student projects.