does the addition of a strong acid increase the percent ionization of a weak acid HB?

Answer 1

No, just the opposite; it decreases it.
HB <===> H^+ + B^-

Ka - (H^+)(B^-)/(HB)
Now if we add a strong acid, such as HCl, notice that is a common ion (the H^+ ion) so by Le Chatelier's Principle, the reaction is shifted to the left (in order to try to undo what has just been done); i.e., H^+ has been added so it tries to remove it and the only way that can happen is for it to move some of the ionized H^+ back to the unionized form. Voila!. More of the unionized form and less of the ionized form means a lower percent ionization.