Would a PhD from a foreign country hurt you in U.S. academia in terms of job offers?
1) I don't know what you mean by second-tier, but TCD isn't at the level of Harvard/MIT/Princeton/Berkeley.

2) The answer does depend on what your range of acceptable careers post-PhD is, because the answer is different for different segments of the job market.  For research positions, it won't matter much, though you have to be careful not to work in an area that is poorly represented in the US.  For primarily teaching-oriented positions (and many US PhDs in mathematics take a teaching position, permanent or temporary, straight after finishing - certainly more than those who take postdocs), you are unlikely to be considered a serious candidate without US (or at least North American) teaching experience.