Was he (Constantine) responsible for the building of St. Peter’s Basilica? And the first ecumenical council at Nicea?
Indeed. He donated several basilicae to the Church to use for worship and personally gave the Church land and finances to build a new basilica as a monument to St. Peter.
And the first ecumenical council at Nicea?
Hosius Cordubensis, a Spanish bishop, proposed the idea - due to the fact that the Arians were gaining ground in Spain.
After consulting with the bishops (who had already held a number of local synods regarding these matters), Constantine offered both a venue and safe passage to to Nicaea for all the bishops able to make the journey. Such a gathering was never possible before - the last time such a Council had been held was in Jerusalem during the time of the Apostles.
Constantine attended, but did not interfere and seated the bishops before he sat down - a breach of royal protocol to demonstrate that he was not in charge: the bishops were.