What happens for DC frequency of a signal when frequency doubler is applied to a signal?
There is nothing like dc frequency. We call the signal which is constant and has no changes in it's value as dc signal and it has zero frequency. The bias signal is one type of dc signal. If we take Fourier representation of a sinusoidal signal like sinusoidal signal [\$\sin(\omega t)\$], it contains the odd harmonics of the fundamental frequency. there is no dc bias present in it. If this signal is applied to frequency doubler, it's frequency would be doubled and consequently the odd harmonics are also doubled. 

Now consider a signal with dc bias like \$A+B\sin(2\pi ft)\$. If we consider the Fourier representation of the signal, it contains a zero frequency component of amplitude (\$A\$) and two frequency components at (\$f\$ and \$-f\$) with amplitude (\$B\$). when this signal is applied to frequency doubler it doesn't have any effect on the zero frequency but the other two frequency components get translated to \$2f\$ and \$-2f\$.

DC bias is entirely independent of the concept of frequency doubling.

Realization of signal using transforms would help to solve the frequency related questions.Hope this answered your question.