Are we allowed to design an auxiliary circuit that contains a bit width over than 2?
for example: Can the circuit b2g3 accept a bus that contains 3 bits?
Yes.
Using multi-bit I/O ports is recommended, over using multiple single-bit I/O ports.
I am not sure you understood the question. I meant is it allowed to use, for example the b2g3 circuit, while implementing the b2g4 (in order to achieve a recursive solution)? since its bit width is over than 2 and according to the instructions that is forbidden
Yes, it is not only allowed but also necessary to use b2g3 in b2g4, because without that you cannot have a recursive solution.
I have updated the instructions for clarity.