A perfect act of contrition only applies to the baptized. One must be a member of the Church to make a perfect act of contrition in the sacramental sense. Just like a man getting ordained. If an unbaptized man is ordained, he is a non-priest, because baptism is the janua sacramentorum, the door of the sacraments. No baptism, no sacramental life, and it does not matter who you are.
No, it does not. Even Old Testament Saints could do acts of perfect contrition. Book of Psalms, 50 :
https://www.drbo.org/chapter/21050.htmThis is a biblical example of perfect contrition.
You are right that baptism is the door to other sacraments and that's the reason it's impossible to ordain a priest who was not baptized. However, martyrdom isn't a sacrament and perfect contrition isn't a sacrament either.