What circumstance led to Mark being outside of Paul's favor? How do we know this problem was resolved? Include scripture references where appropriate.

You can check in Acts 13:5 that John, or Mark, was accompanying Paul and Barnabas. John eventually left them though, as it can be seen in Acts 13:13.

"From Paphos, Paul and his companions sailed to Perga in Pamphylia, where John left them to return to Jerusalem"

When Paul and Barnabas planned another trip, Barnabas wanted John to accompany them, but Paul refused. He did not want someone who left them. Because of that, they went separated ways. Paul went with Silas, Barnabas with Mark.

When Paul wrote a epistle to the Colossians, he included a commendation of Mark in (as you can see in Colossians 4:10). It most likely means that the earlier contention was eventually resolved.