Is it socially acceptable to directly contact renowned academics as a student?

Yes, it's perfectly fine to do so. To maximise the chances of a response, make sure your email is clear, concise and polite.

From my experience, it is very acceptable, and there are very high chances of reply — in my case, reaching 100%.

Highly reputable scientists are people, too, and like to hear comments on their work — of course positive feedback is most welcome, but they are willing to clarify doubts or hear out concerns.

I have successfully contacted authors asking to share their data (no one refused), asking for clarifications, or expressing doubts whether their methodology is bullet-proof. I also repeated some computations and just sent the authors a message saying that it's a nice work and I agree with them. They also replied, and were very pleased with the confirmation of their results.

Some people may direct you to their co-authors or assistants or PhD students for technical details, but that still counts as a helpful reply.

Being an astrophysicist, I was for a moment interested in the IQ distribution, so I contacted, as it turned out, one of the world's most prominent psychologists in the field, who was very polite and helpful: pointed me to semi-public repositories of data and gave advice on how to handle it. In the end, I couldn't contribute anything meaningful, but I would't be discouraged to send him potentially interesting results and propose a collaboration to write a paper together. He made an impression that he'd be very up to it.

As for how-to, start en e-mail with a brief introduction of yourself:

Dear Prof. XXX,

I am a PhD student in YYY at University UUU in WWW. I recently read your work ZZZ and found it very interesting.

Try to keep the message rather short and concise. Scientists are busy and appreciate being precise and getting to the point.

As others have written, yes you may. I would also add some do's and don'ts:

• Don't contact them just to offer praise/thanks (nor derision/dislike).
• Don't contact them to ask for general advice, social commentary, general opinion on something academic (i.e. "What are your thoughts on the theory of XYZ"). Do ask specific and concrete questions (if you have questions).
• Do send email; Don't phone them. If they have office hours at their campus and you're visiting - that's fine too.
• Don't ask for something you could easily find the answer to in your own academic surroundings, or by reading their published work on the subject of your interest. Do look at that published work before writing (but no need to go through dozens of papers and books).
• Don't make them read a very long message explaining context that would be useless to them. Do be concise if possible, and otherwise start with a brief version of what you mean to say, providing a more detailed explanation later.