Juha, in italia nessuno dice "sto faccio la doccia", "sto dormo"! Semmai, se senti qualche romano parlare, dicono "sto a dormì", sto a magnà", ma è dialetto, tutta un'altra storia.... :) Dove hai sentito dire "sto dormo?"

Penso che ti stia confondendo, concordo con la mia connazionale

Good news: Never, never heard of this form! Not even as an error, I believe. So you may just drop it on the floor to be processed when the 'garbage collector' steps in.

"stare + gerundio" is very similar if not identical (as usual am too in a hurry to verify thouroughly) to "I am + ...ing" form in English. As far as I know in English you don't say "I am do my homework" (Io sto faccio i compiti), you say "I am doing my homework" (sto facendo i (miei) compiti) Probably your doubts came from the fact this construction uses "stare" and not "essere". Well, in italian the present progressive is constructed using"stare" not/never using "essere"