Can every CFG be converted into an equivalent CFG of this form?
Since you're allowing $x = \epsilon$ in a production of the form $A \rightarrow xBC$, it seems you can first convert the grammar to Chomsky Normal Form so that every production is of one of the forms $A \rightarrow \epsilon BC$, $A \rightarrow x$, and $A \rightarrow \epsilon$. The only problem is that the productions which produce only a single terminal are not of the form you want, but that isn't too difficult to remedy.