I think that the Romanian use of subjunctive is close to the German sollen in this case (what I'm supposed to do or what should I do).
I mean this sentence translated into German (I'm not a German speaker so please correct me if I'm wrong) could be: Soll ich jetzt oder morgen anfangen?
In fact, "Do I start now?" is perfectly correct gramatically. Nor would I say that it's particularly colloquial. But it has a slighty different meaning from "Shall I".
"Do I start now?" = Is it a fact that I start now?
"Shall I start now?" = Do you want me to start now?