I don't know if I would say that 17 AD is a "not a good guess." Why do you think so? Is it only because Paul refers to Timothy as presently "young" in his letters to him? This could simply be young relative to others, as you said in your first post: Paul, the other disciples, the people in the community where he was teaching/leading, etc. I agree with you that Acts of Timothy is not inspired so we should not draw theological conclusion from it. But documents outside of Scripture are commonly used for historical data (such as dates) and as insight into what people have believed throughout history. But anyways, I agree with you that we can't assume 17 AD is true just because it's in this document. Like I said, I don't think I've seen any studies dating later than 25-30 AD. Here's why...
Please see this timeline of Paul's life:
https://www.blueletterbible.org/study/paul/timeline.cfm
Here's the practical problem I see: Either way you dice it, Paul's first trip to Lystra is just before the Jerusalem Council. You yourself said Acts 15 happens near the start of the establishment of the Gentile churches. Isn't Antioch before this the first known place where a Gentile community of believers was established? Paul picks up Timothy shortly after the Jerusalem Council. All this might be right about the time Galatians is written, but either way Timothy was already traveling with Paul before he wrote all of the canonized letters with the possible exception of maybe one. How was Timothy raised on NT letters when he was already traveling with Paul before Paul started writing the canonized letters, and also shortly after the first known Gentile churches were established and the first documented letter was written to Gentile churches?
Again, this all goes back to when we think he was born. 30 AD is a commonly cited date and this is probably the latest possible date. This would make him about 18 years old when Paul first visited Lystra and about 20 years old when Paul picked him up on his second visit. I find it very difficult to believe that Timothy was much younger than that. What mother is going to let someone take her 10 year old boy, for instance? Do you honestly believe he was this young while accompanying Paul on his early missions? Or being sent on solo missions shortly after that?
Timothy was just seems too old once the church was established in Lystra and he joined Paul a couple years later to have been raised on any NT writings. For those first 18 years or so of his life, reason stands that he would have been taught the Hebrew Scriptures by his mother and grandmother. If not, then he wasn't likely raised on any texts before the establishment of the Lystra church and joining Paul shortly after.
Anyways, I'd really like to see your proposed timeline. Here's an example of one starting with a 30 AD birth. Do you think it's off by many years? If not, that's a lot of childhood years without any NT texts.
30 Timothy born
(Acts 16:1; 2Tim. 1:5; 3:14-15)
47-48
Timothy, Lois and Eunice converted
(Acts 14:8-22; 2Tim. 3:10-12)
- • Paul's first missionary journey
- • Sees Paul (nearly) stoned to death in Lystra
- • Timothy Baptised (1Tim. 6:12)
- • Judaisers preach a “different gospel” – Galatians written
49
Timothy joins Paul on 2[SUP]nd[/SUP] Journey
(Acts 16:1-3)
- • Well reported of by local ecclesias.
- • Prophecies about him given (1Tim. 1:18; 4:14; 2Tim. 1:6)
- • Circumcised by Paul
50
Mission to Thessalonica
(Acts 17:5-15; 1Thess. 3:1-6)
- • Timothy returns to centre of Macedonian persecution
51-52
Timothy with Paul in Corinth for 18 months
(Acts 18:5, 18-22; 1Thess. 1:1; 2Thess. 1:1)
- • Returns from Macedonia with monetary gift (Acts 18:2-5; 2Cor. 11:8-9)
- • Paul goes to Jerusalem. Timothy left behind?
52-55
Timothy goes to Corinth
(Acts 19; 1Cor 1:1; 4:15-17; 16:10-11; 2Cor. 1:1)
- • Timothy with Paul in Ephesus for 3 years
- • Sent to Corinth (and Macedonia too) (Acts 19:22)
56-57
Accompanies Paul to Macedonia and Greece, and Jerusalem
(Acts 20:1-6; Rom. 16:21)
60-62
With Paul in Rome and Philippian Mission
(Acts 28:16-31; Phil. 1:1; 2:19-23; Col. 1:1; Phm. 1:1)
63-64
Timothy's Ephesian Mission
(1Tim. 1:3; 3:14-15; 4:12-16; 6:20-21)
- • Paul released, and revisits Asia Minor, Macedonia, Crete (1Tim.1:3; Tit. 1:5)
65
Timothy goes to Rome, Paul executed
(2Tim. 4:6, 9-21)
65?
Timothy imprisoned and released
(Heb. 13:23)