I see this is a repost. I was too busy to give you an answer the other day.

To deal with your specific questions first. Yes, there was a clergy at the time of becoming the State Religion under Theodosius I in 380. In fact, by this time there was a fairly developed ecclesiastical order. Bishops existed in most major urban areas, and exercised 'monarchical' episcopacy in those regions - that is a single bishop had authority over the city and its surrounding area. The bishop would then have a number of presbyters under them, and deacons involved in ministerial duties, and then a number of other 'orders' such as readers, etc..

Priests (= presbyters) were trained 'in house' so to speak, through that hierarchy system, and more in what I'd call an 'apprentice-type' relationship to senior clergy. But in the 4th century most of those who became significant bishops / leaders had an education in the classical rhetoric tradition before becoming church leaders, and often before becoming Christians altogether. Rhetorical training was higher education. There were not, really, anything resembling modern seminaries.

Many elements of what would become modern Catholicism already existed. A fixed liturgy, sunday worship, baptism and eucharistic rites, etc.. I'm happy to elaborate on anything in particular.