Timothy was half Jewish, was his mother or Father Jewish?

It was his mother who was Jewish. Even today the child of a Jewish mother is considered Jewish whether her husband is Jew or Gentile. The child of a Jewish father and Gentile mother is not considered Jewish and must go through a conversion.

Source: Acts 16:1

A relative of Barnabas I traveled with Paul on his 1st missionary Journey. We had a falling out but later he asked Timothy to bring me to Rome to help him. Who am I?

I am Mark or more properly John Mark. I was an early missionary of the church. Though some dispute it I am also credited with writing The Gospel Of Mark.

Source:  Acts 15:36-39 and 2 Timothy 4:11  (Corrected with thanks to Gary.)

Titus was bishop of Crete, Timothy was appointed Bishop of what region?

He was Bishop of Ephesus. Paul used Timothy and Titus and others to nurture the churches he planted.

Source: 2 Timothy 4:22

Timothy was Paul’s assistant and son in the faith. Where was he from?

He could be from Lystra or Derbe, it is generally accepted that he is from Lystra.

Source: Acts 16:1

In which books of the bible is Titus mentioned by name?

Titus was mentioned in the following books; 2 Corinthians, Galatians, 2 Timothy, and Titus.

Source: See Above