I have been hearing for a very long time from many people who call themselves Christian that people should not have sex before Marriage
If they didn't say that they couldn't legitimately call themselves Christians.
yet most of them do not seem to understand Marriage is much more than just abstaining from sex before Marriage
They don't understand that it's more than that? Could you explain?
God gave us Marriage with the sole purpose of bringing Children into the world to train them them in Catholic faith in order to populate heaven.
That's right. Can you tell us who has been saying otherwise?
Can a man and a woman have a valid Marriage if they did not receive the Catholic Sacrament of marriage?
Short answer, yes. However, baptized Catholics are required to marry as Catholics.
Is order to receive the Sacrament of Marriage does the Traditional Catholic Mass have to be said within the ceremony?
Catholic marriage does not have to take place in the context of mass.
Can a Protestant perform the Catholic Sacrament of Marriage?
In Catholic theology, the form and matter of the sacrament does not depend on the priest. However, to have a valid marriage, usually it is required to have the cooperation of the priest with proper jurisdiction.
Can someone who got married by the justice of peace claim to not have a valid marriage and then receive the Sacrament of Marriage in a Traditional Catholic Church with a new partner?
That would depend on many things. You would need to talk to a priest about. Civil marriages of baptized Catholics were not traditionally valid.
Can a man and a woman have a strong, intimate and loving relationship without the Sacrament of Marriage?
Can non-Christians have strong, intimate and loving relationships? I suppose it is possible in some sense, since they have a natural morality engraved in their hearts. However, someone who is a Catholic should marry another Catholic in a sacramental marriage.
If someone comes to the knowledge they are in an invalid marriage with illegitimate children, what is their moral duty in this situation?
You need to talk to a traditional priest about this. Children of annulled marriages are not considered illegitimate.
I am not Married but as the moral fabric of society falls apart I have been asking myself, “What does God ordain for a valid Marriage to take place?"
That's a question to ask a traditional priest versed in canon law. But if there were no clear impediments, not the subjective nonsense that's used today to "annull" a marriage, but the impediments that were traditionally held to be impediments, then you should presume that the marriage was valid.