You said:
Dear Sir/Madam,

I am a single Arabic Catholic lady who is happy being Catholic. I love my religion so much and I do not wish to make any mistakes that would depart from my faith. That's why I hope you can assist me. I have a few questions I hope you can answer soon:

We're happy to answer your questions. By the way, I go to a Melkite Catholic church myself which is predominantly Arab.

You said:

• Are we allowed to have sex before we get married?

Of course, I mean get married in the Church. This question came to mind because a friend, Ed, and my family have taught me that we should get our marriage blessed from God first — meaning we should get married in the Church, then have sex and, God willing, bring forth a new baby.

• If we had sex before our marriage is that a big sin?

Sex is an expression of the total life-giving gift of self between man and woman — it means,
I give everything to you: my life, my body, the most intimate reaches of my being. Because of the radical nature of the gift, God designed it only to be given in the context of a life-long vow of marriage, which sex expresses. After all, you wouldn't want to give yourself completely to someone who might run off tomorrow.

Sex apart from this life-long public commitment — i.e. marriage — is called fornication and is strictly forbidden by the Bible and by the teachings of Christ, and hence by the Church (whose role it is to communicate the teachings of Christ). When you have sex before marriage, you are not giving yourself totally to that person since you have not made a public vow of fidelity to them.
You are holding yourself back. Because sex expresses with your body your total self-giving,
you are living a lie.

You said:
I have one Muslim man who would like to become a Christian.

• Is that's allowed?
• What does he have to do?
• Is he allowed to get married to a Christian lady in the Catholic Church?

Of course, a Muslim can become Christian! It has some potentially serious repercussions on the Muslim side, but from the Christian side we would welcome him.

He needs to go to the local Catholic priest and indicate he wants to be baptized and become Catholic. He will receive instruction and then be baptized into the Church. Whether he becomes a Catholic (or Christian) or not, he can still marry a Christian lady, though if both are Catholic,
it makes for a better and more stable marriage and family.

Obviously one of them has to be Catholic to get married in the Catholic Church.

Check out How do I become Catholic and related questions.