I am happy that I don´t know anybody who likes a synthetic skin on a bodhrán ! The only way for a trad instrument is a good auld goat skin, my opinion. I only use goat skin. Most common size is 16" and 5,5" deep. I only make tuneable ones.
I agree about the goatskin. I have 4 bodhran's, 2 tuneable and 2 non tuneable, all goatskin. The nontuneables have a place in my heart, because that is how the drum was made originally and I like their sound and personality.
I don´t know if there is a name for the hand style. My friend Johnny "Ringo" McDonagh (yesssss I want to show off ;o)))))))))) ) is doing a lot of hand style as well. I´ll ask him some day. I am not sure, but I a think the Limerick style is with a short beater that is fixed on the thumb with a leather string. You use it in a kind of bell mallet in a way of topendstyle becuase you only use one end of the stick. As I said before: I am not sure.