Would this be used to say he has new numbers as in He has a new telephone number? or He has new numbers to show in an accounting or sales chart?
It could be used to refer to multiple telephone numbers; I don’t know if the “financial information” meaning could also be translated by uimhreacha.
Does the illicit gambling known as "the numbers game" exist in Ireland?
No. Gambling is not as heavily regulated outside the US, so it not as lucrative for criminal involvement (that's not to say that there so no such thing as illegal gambling, but it doesn't involved the small money that "the numbers game" encompasses). The phrase "a numbers game" only carries the connotation of who can drum up the most support.