Was Christianity made the official language of Rome in AD 337?

Answer 1

I believe you mean "religion", not "language". Answer is No. Theodosius I declared the "Catholic Church" to be the only legitimate Imperial religion on 27 Feb, 380.

An earlier Roman emperor, Constantine, had declared himself to be a Christian around 313. In 325 he summoned the Council of Nicaea, effectively the first Ecumenical Council -- clearly an indication of his growing role in the Christian world. He was finally baptized in 337, just before his death. This settled any doubt, and he was certainly the first Roman emperor to convert to Christianity. However, he did not declare Christianity to be the official religion of Rome. That task was left for Theodosius I.