First question: No, Jesus told him that someone will tell him what he must do
5 And he said, Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest: it is hard for thee to kick against the pricks.
6 And he trembling and astonished said, Lord, what wilt thou have me to do? And the Lord said unto him, Arise, and go into the city, and it shall be told thee what thou must do.
Second question:
Usually when one begins to believe, they are lead to repent, confess and be baptized.
In Acts 2 the people were cut to the heart. They now believed in peters message. "What do we do?" they asked. Peter told them Acts 2:38.
Third question:
Cornelius went through the same situation basically. when he got the Holy Spirit it tells us it was for miraculous gifts ("scales fell off his eyes" Paul) Cornelius spoke in tongues.
Still, to obey the word of God, one must be baptized to be saved.
Cornelius did it "who can forbid them water... and peter commanded them to be baptized"
Paul did it "Arise, why are you waiting, be baptized and wash away your sins calling on the name of the Lord."
By the way, baptism is for the remission of sins too. Hence the wash away your sins Acts 2:38; Titus 3:5; 1 Peter 3:20-21