Yes - the existence of a left-invariant metric on any Hausdorff first countable group is the Birkhoff-Kakutani theorem. Given a left-invariant metric $d$ on a compact group then just put $$ d'(a,b)=\sup_x d(ax,bx) \;. $$

Yes - the existence of a left-invariant metric on any Hausdorff first countable group is the Birkhoff-Kakutani theorem. Given a left-invariant metric $d$ on a compact group then just put $$ d'(a,b)=\sup_x d(ax,bx) \;. $$

PS Of course, the point is that all these metrics metrize the group topology.