Brute-forcing Symmetric Keys
No, it doesn't help.  It doesn't hurt either; as long as you don't repeat keys, the probability of success is always the same.  That is, if there are $2^n$ possible keys, and you test $\lambda$ of them, the probability you hit the right one is always $\lambda / 2^{n}$.

A key generated by a high quality random number generator (or a good key derivation function) is equidistributed; every possible value has an equal probability of being the correct one.  Yes, most keys will have approximately the same number of 0's and 1's; but that's because most possible values will have approximately the same number of 0's and 1's.