Is Limerick, Ireland in any way connected with the limericks that people speak?

The origin of the actual word limerick is obscure. The first known occurrence is from May 1896; the OED first reports it in 1898. The name is often linked to an earlier form of nonsense verse which was traditionally followed by the refrain that ended "鈥ome all the way up to Limerick?", Limerick being an Irish city. That the older refrain does not match the meter of the limerick has been used to attack this theory. A point in favour, however, is the fact that in other languages, limericks are indeed sung, with wordless (la-la) refrains between them that match a version of this text.

Yes it is. In fact there is a bar in the castle in Limerick where the wall is plastered with Limericks.