PARIS — If Rachida Dati realizes her long-held dream of becoming Paris mayor in the coming days, her victory will represent the culmination of a remarkable journey from her childhood in an impoverished social housing complex to the gilded chambers of City Hall. The pugnacious former culture minister from the conservative Les Républicains party is running neck-and-neck with the Socialist Emmanuel Grégoire in polls running up to municipal elections on March 15 and 22. A disciple of former right-