Might Nimrod have had something to do with the building of the city and tower of Babel (Gen 11:3-4)?

Yes, and in fact this is tolerably clear from the text, considering the background on him given in the previous chapter (Gen 10:8-12). In that section, we were told that “the beginning of his kingdom was Babel” (10:10). And here we have the first listed city—and one specially distinguished by being discussed at such length. In the same way that Nimrod is singled out for his earthly accomplishments, Babel is singled out. So it seems likely, even if it is not stated in so many words, that the person behind the building of the Tower and the City of Babel described in Gen 11 was Nimrod. Moreover, given the length of years of lifetimes, it is wholly plausible that Shem’s great-grandson Peleg would have been present, a man in his prime, at the same time that Ham’s grandson Nimrod made a name for himself as a “mighty hunter” and, it seems, not just a king but the founder of Babylon.