May a cohen choose a shiduch without inquiarying in full about potential past relations with a mamzer or a gentile? Would it be permitted to a cohen to tell a girl before becoming engaged never to discover to him she lost her virginity (and to a gentile in particular)? May an old cohen begin looking for a match and meeting girls while still in the middle of the process of gerushin?

Ideally a cohen should be as scrupulous in regard to a shidduch as one would be with eating kosher food, and ask for a "hechsher" that there is no potential conflict. In special curcumstances there might be room to consult with a qualified posek about forgoing some inquiries. I don't understand why the last question is about a cohen. Generally it is improper to be involved in a future shidduch while still married, but I can imagine curcumstances in which it would be proper. Again, I advise consult on specifics.