Is my second marriage legitimate and pleasing to Allah and His Messenger, and if not, what should we do?

For a marriage to be valid, it must be contracted by the woman's guardian (wali). If your paternal uncle prays, he is your guardian. Otherwise, guardianship transfers to the next in line. The maternal uncle is not a guardian in marriage unless he is deputized by the guardian or a Sharia judge rules the contract valid. Your first marriage is valid if the judicial system in your country rules valid a woman marrying herself. The second marriage is not valid because it was without a guardian and was not ruled upon by a Sharia judge; its ruling is annulment. If a divorce occurred based on absolution, it is a khulʿ. If you had menstruated once before your third marriage, it is valid; otherwise, it is not.