The impeachment has legal basis on administrative improbity. This is what the law says:
"Acts of the President against the Constitution, and, especially, against the guard and legal use of public money".
The definition of "against the guard and legal use of public money" is given as, among other things,
"neglect the collection of rents, taxes and fees, as well as the conservation of national patrimony."
Futhermore:
"Are crimes of liability against the probity of administration: (...) Not make effective the responsibility of his subordinates, considered manifested in functional offenses or the practice of acts contrary to the Constitution."
and
"Constitutes an act of improper conduct which infringes upon the principles of government action or omission that violates the duties of honesty, impartiality, legality and loyalty to the institutions"
I'm not sure what you mean with your fire alarm allegory, but I suppose you're trying to say that all the investigations are only happening because the government (and the President in particular) let them happen. This is false. The Federal Police and the Public Attorney's Office are independent of the government and do not require it to allow any kind of investigation.
Edit: by the way, weather the impeachment is legal or not is irrelevant to the point of weather the protesters are representative of Brazil's population (they are). You seemed to imply that somehow due to who organized part of the protests, they are not, but then nothing in your latest reply addresses that.