Can I compress a 6 inch (R-20) batt to fit into a 2″x 4″ cavity? Will it be more effective than using a 4 inch (R12) batt?

While insulation batts are easily compressed, the R-value will be significantly reduced – it is best to use the correct thickness and not compress the insulation. You should also avoid storing heavy materials on top of attic insulation for the same reason.