Is there a rule for dropping the definite article in front of a
e.g. "Aqui está a sua chave. / Aqui está sua chave."
I'm espacially interested how Brazilians handle this.
Thanks in advance
Small correction: what you call a demonstrative is actually a possessive. Demonstratives are words such as "esse", "este", "aquele", etc., which correspond to English "that" and "this".
Both forms you mention are considered correct in Standard Portuguese. As far as Brazilian Portuguese is concerned, I would say that, as a rule of thumb, including the article is more common in colloquial speech, while omitting it is preferred in a more formal setting. Even so, this "rule" is just a guideline, and by no means absolute; it is possible to find examples of both forms in both settings. Thus, the Brazilian constitution has several occurrences of the possessives "seu"/"sua", but only a few of them are preceded by an article. This is similar to how some authors prefer using non-abbreviated forms such as "does not" and "are not" in writing, as opposed to abbreviated ones such as "doesn't", "aren't", etc., even if both forms are accepted in Standard English.
Edit: It is worth pointing out that these words ("meu", "seu", etc.) can also serve as pronouns; in that case, the use of the article is mandatory. For example:
etc. This usage corresponds to words such as "mine", "theirs", "ours", etc. in English.
At a guess, the article is often joined to the verb (está + a) in the spoken language, so much so that we don't always hear it. Let's use a masculine object so as not to have that junction, and I think most people would include the article ("aqui está o seu carro"). Without the article the sentence wouldn't be wrong, but in the written language I would always use it.
Conclusion: in the spoken language, in Brazil, nobody cares whether you say "sua chave" or "a sua chave".