Does the Church recognize common law marriages as valid? Or are the common law spouses in a state of sin?
If a country/state permits common law marriage, the Church would presume such marriages valid for eligible couples, unless proven otherwise.
Common law marriages, if permitted in civil law, have all the rights and privileges as marriages witnessed by an officiant.
If a Catholic got proper permission/dispensation from their bishop, they could also receive the civil benefits of marriage through the common law.
Could a common law marriage that is recognized as such by civil authorities be subsequently convalidated in the Church?
Common Law marriages are, by definition, recognized by civil authorities.
However, such a mode of marriage must be permitted under civil law. In the US, many states prohibit common law marriages in their boundaries. A couple
cannot be married through the common law if the state abolished common law marriage (though a common law marriage formed in another state would be fully recognized in most cases).
Of course, a common law marriage must be entered in good faith.