I know one can say [hundreds digit] + "cento" for centuries from the twelveth to the twentieth, e.g. "Quattrocento" for the fifteenth century. But if I want to say "Ventunesimo secolo", can I say "il Diecicento"?
No, you can't say “diecicento”. The expression “il Quattrocento” is short for “il Millequattrocento”, meaning the fifteenth century (or, more precisely, the years starting from 1400 to 1499). Usually, “il Quattrocento”, “il Cinquecento” and so on are referred to the artistic or historical peculiarities of the century: for instance, *il barocco è tipico del Seicento, ma si estende anche nel primo Settecento”.
Note that “Quattrocento” is much shorter than “il quindicesimo secolo”, one of the reasons the former may be preferred.
I don't think anybody has ever said “il Cento” for the twelfth century. Anyway, the idea is that the initial “mille” is dropped; you can't drop “duemila”, at least for the time being.