During the Pax Romana, travel was possible, and missionaries (who spread Christianity to non-believers) felt comfortable and safe when using the roads, and could spread their message without fear of attack. There were open routes throughout the Empire, and people were more interested in learning the teachings of Christ during a time of peace than they would be during a war or famine- it was big news. Pretty much everyone spoke the same language at the time, as well, so the missionaries could preach easily to the people.
There are a lot of periods in history that influenced the growth of the religion. In my opinion, probably one of the most important is when Constantine the Great converted to Christianity, making it the Roman Empire's main religion, converting the citizens from their Pagan ways to Christians. And of course the Romans like to conquer, and with conquering, they spread their religion, so thats your answer :)
Yup, but I think christianity has some like-able qualities in itself. One is that the religion applies to everyone, even the poor and weak. So during the Roman Empire, I would think the reason the rulers didn't like the religion is because it gave the poorer people hope and also gave them reason not to worship or look up to the rulers in the same way. The fundamental message of love is a nice one, so I think Christianity in itself is a likable religion, which helped it become more popular as well