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There is absolutely no requirement for Catholics to ever pray the Rosary, or to ever say a "Hail Mary". Who told you there were such requirements for Catholics? (If you want to know what Catholics really teach, read the "Catechism of the Catholic Church".)
(However, why would someone not want Mary (or any other saint) to pray for them? I want as many people praying for me as possible, and it would seem senseless to me to not seek those prayers of the saints and others when there is a need.)
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"Suppose a man prays to Jesus for His salvation and dies unbaptized. Will he be immediatly in God’s presence?"
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Nothing is impossible with God. Think about the so-called "good thief" on the cross. As far as we know, there is no record of him having ever been baptized. God uses ordinary means to do things, and sometimes uses extraordinary means to do things. My belief is that we should learn and do whatever we can to determine and carry out what God wants us to do (to the best of our ability), but we should also always trust fully in God and His loving providence.
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I come across EWTN a few times a week. They always seem to be saying the rosary. Why do they do that if it is not important?
Another poster cited The Necessary Seven Sacraments.
So, the “good thief” was obviously a sinner. Was his faith in Christ and Jesus' sacrifice sufficient to get him straight into Heaven or did he need a some “therapeutic” Purgatory torture?