Are both electrons and holes responsible for conductivity in an extrinsic semiconductor (both n-type and p-type)?
Yes, electrical conductivity is a function of both electron and hole concentrations, in all semiconductors, intrinsic or extrinsic. 

$$\sigma=\mu_nn+\mu_pp$$

However, since the electron and hole concentrations have an inverse relationship, unless your semiconductor is very lightly doped or intrinsic, the vast majority of conductivity will be from either electrons or holes, whichever has the higher concentration, the majority carriers.