For beloved see 1 Tim 6.2. The term is based on the Greek word for love (agapē) and is therefore an expression of Paul’s intimate endearment to Timothy. Another way of expressing To Timothy, my beloved child is “Timothy, you are like a dear child to me.”

On the translation of Grace, mercy, and peace from God the Father and Christ Jesus our Lord, see 1 Tim 1.2b.

Alternative translation models for verses 1-2a are:
• I, Paul, who am an apostle (or, personal representative) of Christ Jesus, write this letter to you, Timothy. God chose me to be an apostle, and he gave me the life which he had promised to give me. This life comes as a result of being in a close relationship with Christ Jesus.
• Timothy, you are like a dear child to me.
I pray that God, who is our [inclusive] father, and Christ Jesus, the one who commands us [inclusive], be kind to you and cause you to live in peace.

Or:
• … God chose me to be an apostle so that I might proclaim the life which he promised to give me. This life comes….

Quoted with permission from Arichea, Daniel C. and Hatton, Howard A. A Handbook on Paul’s Second Letter to Timothy. (UBS Handbook Series). New York: UBS, 1995. For this and other handbooks for translators see here .