Does Gibbs free energy of formation always equal zero for elements in standard state?
Does Gibbs free energy of formation always equal zero for elements in standard state?


No.  Elements occur in different allotropes.  

$\Delta H_f^o$ and $\Delta G_f^o$ are defined to be zero at 298K, 1 bar for the lowest energy allotrope, with the exception that the values for white phosphorous are defined to be zero even though it is not the lowest energy allotrope.  

The other allotropes do not have zero $\Delta H_f^o$ and $\Delta G_f^o$ even though they are still elements.  


  So of course the compounds that have a $\Delta H_f$ of zero are the elements in their standard states but why does that make their $\Delta G_f$ zero?


$\Delta H_f = 0$ does not imply $\Delta G_f = 0$

The two values are independently defined to be zero for a particular allotrope as explained above.  


  Does every compound that has a $\Delta H_f$ of zero also have a $\Delta G_f$ of zero?


If a compound (other than by definition) happens to have $\Delta H_f = 0$, then $\Delta G_f$ could be any value, it is not necessarily zero.  


  What exactly is $\Delta G_f$ 


It is the change in Gibbs energy upon forming a compound in its standard state from its constituent elements in their standard states at 298K.