I can answer the above question for the Slovak airmen, nominaly those of Letka 13. Officially they remained the part of Slovakian armed forces, wore their own country uniforms, had their own squadron comanding oficer etc. The problem was that Slovakia didn't have enough modern aircraft for successful deployment on the front. So the Germans loaned their own aircraft Bf 109F G to Slovakian unit and the unit was organisationaly made part of JG 52 receiveing their orders from Stab of III/JG 52.