All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
First, you were mistaken for acting in this manner. You should have offered the matter of your second marriage to your guardian if you had one. Your half-brother is eligible to be your guardian if he is your father's son. If he is your mother's son, however, he is not eligible to be your guardian. In this case, the judge becomes your guardian. You are required to adhere to the Ruling of Allaah even if your ex-husband will take the children.
Here are the answers to your questions:
The answer to the first question:
If the marriage official who concluded the marital contract believes that the marriage is valid without a guardian, according to the Hanafi School of Fiqh, and further he is knowledgeable and you follow his opinion, then your marriage is valid. But if he does not believe in that and he concluded the contract without a guardian because there was no guardian for you in this country, then for the marital contract to be valid, the following conditions should be fulfilled:
a. The marriage official should be a judge and officially authorized to conclude marriage contracts.
b. The contract should be concluded in the city where that man is authorized to do his job.
c. The contract must be concluded in the presence of the woman.
d. The woman has no eligible guardian in the judge's country.
e. The distance between the woman and her guardian should not be less than 85 kilometers.
When these conditions are fulfilled, the marital contract is valid. Otherwise, it is invalid.
The answer to the second question:
You are required only to sincerely repent to Allaah and regret your past deeds.
The answer to the third question:
Whether your second marriage is valid or not, you are required to observe the ‘Iddah. It is three times of purity for those who experience menstruation and three months for those who do not. There is a possibility that intercourse occurred mistakenly out of belief that it is lawful.
The answer to the fourth question:
You are not permitted to remarry your children's father unless you observe the ‘Iddah after divorce from your second husband.
Allaah Knows best.