Janh:
My first marriage was performed by a minister at city hall. I was unable to conceive, and we adopted two children. Does the Catholic church consider this first marriage valid?
If you and your first husband were both baptized Christians, then your marriage is presumed to be valid and Sacramental.
If one or both of you were not baptized, and remained unbaptized, then you had what is called a “good and natural” marriage. This is not the same as a Sacramental marriage, but it does not mean you were free to divorce and remarry either.
To determine the validity of your first marriage and whether you were free to contract another marriage, your first marriage would have to be examined by a Catholic marriage tribunal.
Janh:
My first husband and I divorced, I married a Catholic (in a private ceremony) who was divorced by his wife, and we have had a baby. How does the church view a man whose wife divorced him?
Not enough information. Again, if both were baptized, and presuming the Catholic followed the required form of marriage then the marriage is presumed to be valid and Sacramental. The fact that his wife left him is not relevant to the Sacramentality of the marriage and whether he would be free to contract another marriage. His marriage must be examined by the tribunal.
Janh:
Are there any circumstances under which we could have a wedding in the Catholic church?
Both of your prior marriages need to be examined by the Catholic marriage tribunal. If both are found to be invalid, then and only then would you be free to marry each other in the Church. At this time, you are both in a state of adultery.
Your first step should be to make an appointment with a priest in your local area, talk to him about the details, and then get the ball rolling to have both marriages examined. The priest can guide you.