You're probably looking for the smallest nonzero binary number, which would depend on the number of bits allowed and the representation format. For example, if you're using IEEE 754-1985, the smallest single-precision (32-bit) number is ±1.18×10^(-38), and the smallest double-precision (64-bit) number is ±2.23×10^(-308).

I'd say zero, unless you allowed negative numbers. In complement representation it would be limited by the number of bits available. If you just mean base two, you could write an arbitrarily small number, your limits would be storing it. (For example, after writing -1111111111111111111111...... followed by a trillion 1's you might start running out of paper to write on.)

There is almost no limit because you can design your own binary code.

0001 = Holden

0010 = EddieJ

0011 = Obama

0100 = Bart Simpson

etc