The Cohen/Levi thing still is used in the Torah service, but in day-to-day practice, all Jews are seen as equal... the Cohanim and Levites just have a few extra responsibilities on Saturday morning.
Converts are regarded as full Jews, whether you're talking about Orthodox or Reform (or Masorti or Conservative or Liberal). There is no caste system in Judaism. In Orthodox communities, Cohens and Levites are still seen as having particular roles but apart from certain tasks within the synagogue, there is nothing to differentiate them from any other Jew. In Reform these minimal differences are not observed.