May a Cohen marry a young girl, who, when two years old, had been adopted by orthodox childless parents, and immediatly converted Kadat Vekadin? (her biological parents and family are unknown)

According to both the Rambam (Hilchot Issurei Biah 18;3) and Shulchan Aruch (Even HaEzer 6;8) a Cohen may not marry a Gioret (convert) even if the conversion took place at a very early age (below three). In the case you describe the assumption is that the child was not born Jewish. If the conversion took place in a city where most of the population is Jewish, and the conversion was done just in case, a competent Beit Din would have to discuss the exact status of the child in terms of marrying a Cohen. (See Responsa “Mishne Halachot” of Rabbi Menashe Kline vol. 14 ch. 18) In a case where there is already an engagement or even a serious relationship, a Beit Din should certainly be consulted.