Is the marriage contract considered valid if it was concluded without a guardian (wali), but with witnesses and the knowledge and consent of the family, after the wife had previously lived with the husband without marriage and lost her virginity?

This contract is void for two reasons: First, it was done without a guardian (wali), and second, it occurred before istibra (cleansing of the womb) from adultery.

You both must repent to Allah for the adultery. If you repent and you (the woman) wait until your istibra is complete, then you can remarry with the fulfillment of the conditions of marriage, including a guardian, a dowry (mahr), and witnesses.