Burton L. Mack (Who Wrote the New Testament) says that the attribution to Paul of the epistles to Timothy (1 Timothy, 2 Timothy) is clearly fictional, for their language, style and thought are thoroughly un-Pauline. This is the consensus view of most New Testament scholars.
The two letters, as well as the Epistle to Titus were written at different times, undoubtedly during the first half of the second century, as demonstrated by references to the role of bishops, who begin to appear early in the second century. Their content fits nicely into the situation and thought of the church in the mid-second century. The epistles were not included in Marcion's list of Paul's letters (ca 140 CE), and quotations first appear in lrenaeus' Against Heresies, dated about 180 CE.
We do not know who wrote First Timothy, or just when, but it was written some time in the first half of the second century.
2 Timothy 1:2 refers to Timothy as Paul's dearly beloved son, suggesting that Timothy was probably a young man.
However, Second Timothy was not really written to Timothy or, for that matter, written by Paul himself. Nearly all New Testament scholars say that the epistles to Timothy and Titus, together known as the Pastoral epistles, were written during the second century in order to deal with pastoral issues that were beginning to arise in the Church as it expanded. The author realised that by using Paul's name, the epistle would gain far greater acceptance than if he had acknowledged his own role as the author. Both Paul and Timothy were dead when the epistle was actually written, so we can not really say how old Timothy was.
As the questioner asks 'where' was Timothy (in Ephesus, circa 62 AD), let Scripture reply:
1 Timothy 1-3 New International Version (NIV)
1 Timothy 1
1 Paul, an apostle of Christ Jesus by the command of God our Savior and of Christ Jesus our hope,
2 To Timothy my true son in the faith:
Grace, mercy and peace from God the Father and Christ Jesus our Lord.
Timothy Charged to Oppose False Teachers3 As I urged you when I went into Macedonia, stay there in Ephesus so that you may command certain people not to teach false doctrines any longer 4 or to devote themselves to myths and endless genealogies.
The overwhelming consensus of biblical scholars is that 1 Timothy was written in Paul's name during the second century, to address problems that had not yet arisen during the lifetime of Paul. It was not really an epistle to Timothy, but an encyclical to the Church as a whole, using the name of Timothy to add authenticity to the epistle. Just as Paul was long dead at the time 1 Timothy was written, so was Timothy.
I don't know,but that's what help him become a "SAINT"
Timothy Leary died on May 31, 1996 at the age of 75.
Timothy McVeigh died on June 11, 2001 at the age of 33.
Timothy Treadwell died on October 5, 2003 at the age of 46.
Timothy Carey died on May 11, 1994 at the age of 65.
Timothy Beaumont died on April 8, 2008 at the age of 79.
Timothy Ruggles died on August 4, 1795 at the age of 83.
Timothy Birdsall died on June 10, 1963 at the age of 27.
St. Benezet was 21 years old when he died.
St. Timothy's School was created in 1882.
St. Clare of Assisi was 61 years old when she died.
Timothy Treadwell was born on April 29, 1957 and died on October 5, 2003. Timothy Treadwell would have been 46 years old at the time of death or 58 years old today.
Timothy Carey was born on March 11, 1929 and died on May 11, 1994. Timothy Carey would have been 65 years old at the time of death or 86 years old today.