Yes it can, see the paper A new proof of the Ahlfors five islands theorem by Walter Bergweiler. The idea is to first prove the result where the Jordan domains are disks (which is much more elementary and essentially only uses a rescaling lemma for normal families) and then show that the general case can be reduced to the case of disks. The second part uses the existence theorem for solutions of the Beltrami equation (so-called Measurable Riemann mapping theorem) but other than that is not too difficult.

You could also take a look at Bergweiler's other paper The role of the Ahlfors five islands theorem in complex dynamics for interesting applications.

Yes it can, see the paper A new proof of the Ahlfors five islands theorem by Walter Bergweiler. The proof is based on a result on Nevanlinna concerning perfectly branched values, Zalcman's rescaling lemma for non-normal families and the existence of solutions to the Beltrami equation (so-called Measurable Riemann mapping theorem).

You could also take a look at Bergweiler's other paper The role of the Ahlfors five islands theorem in complex dynamics for interesting applications.