Marriage

Question

1. Can a kohen marry a girl who was adopted and converted as a baby? 2. If I am dating a girl (baalat teshuvah) and I find out that her mother was divorced before marrying her second husband and giving birth to her, can I assume that the mother's first marriage was not halakhic, or am I obligated to ask some questions, and perhaps discover that the girl is a mamzeret? 3. If a couple becomes religious, do we ask them to go through another marriage ceremony, or do we assume that living together as husband and wife created a kosher marriage (if yes, does this affect the answer to no. 2, or is it that we decide le-kula in both cases)?