It would mean a very big grin. The moon of May is an expression that features more than once in Playboy of the Western World. The play is Irish and full of colloquialisms. From what I can work out, moon of May seems to represent an extreme. It may be that it refers to a full moon but that's only a guess. Someone with more knowledge of Irish colloquialisms may have a better idea.
Hi tarama, "grin your ears off" is not an established expression. A search on Google only threw up eight results for me, five of them from the same source material. However it is easily understood by analogy with other established expressions like "laugh your head off" and "cry your eyes out". In other words, the action (grinning, laughing, crying) is taken to the extreme.