What should I use to seal paper mache to make it strong?

Epoxy is probably your best bet. It is very durable and will harden nicely around your mask. Epoxy also protects against the elements well and it should be fine to take your mask outside. I'm not sure sure about the paint, but acrylic might work well over the epoxy. You should also consult an associate for advice at whatever store you will buy the epoxy at. I hope that this helped!