Would submitting an improved, but after-the-deadline, personal statement be seen negatively by the adms committee? On one hand it´d be a better crafted, with more relevant content, and more attractive to read piece of writing; on the other hand, it´d be, well, after the school´s deadline. In your experience, would it really help?

I think it would not help. I think it would rather harm and be seen as very negative, or at least create such an impression.

Deadlines are not just ballpark figures-they do actually have a purpose.

Only thing to update would be your contact details, or if you have received an important degree result.

What about submitting a writing sample, such as a monography on one of the research interests mentioned in the statement?

Why?

How about just leaving as it is and to wait for result/feedback/admission or whatever you are waiting for. In this stage you don't want to create unnecessary attention (which could be counter-productive), you want to keep you head down, keep a low profile and wait for the result.

It happens that nowhere on the school´s website it is mentioned that applicants may submit writing samples, but someone from there just told me that "some applicants send thesis, conference papers, etc".

I think it makes sense that if an applicant sends a research sample in one of the fields of a key professor, her/his chances will improve even if this is post-deadline (after all, they don´t mention it at all in their application guidelines).

That depends on the quality of the submitted work, if it's well-written, provides many references of the potential supervisor's work, provides a similar theoretical position etc.

My opinion is that it's a mine field and could do potentially do a lot of harm to submit some past work, in particular if it's not explicitly asked for.

Unless of course, you have published widely in top journals and/or won the Nobel prize. Although the latter two could also diminish your chances of getting accepted as the magic rule in academia is:

"Never outshine your master"

(But don't be a fool, either)

Lol, that was a good one.

I see your point. I was thinking about submitting a 2-3 thousand essay on one of this prominent professor´s field of research, but it´d not contain data analysis, modeling, etc, only literature survey.

Would you say this is the riskiest thing to do?, given that with surveys it´s more complicated to attain some "originality level" necessary to appropriately impress the adms committee.

was the essay you want to send them recently published or do you just want to write it up now for the purpose of submitting it with your application? if it was published I would send a copy of the original paper and state in not to many words that it was submitted prior to your application and is now published. Otherwise I agree with the other: wait and see. With sending things after the deadline it could make the impression that: you did not take your time before, your were not organised enough to send everything, you did not take this one serious but now got declined another one and want to make up...