I have a comment on one online question about whether Timothy’s father was a believer or not. Philippians 2:22 seems to indicate that Timothy’s father was a believer and co-worker with Paul and Timothy.
"But I hope in the Lord Jesus to send Timothy to you shortly, so that I also may be encouraged when I learn of your condition. For I have no one else of kindred spirit who will genuinely be concerned for your welfare. For they all seek after their own interests, not those of Christ Jesus. But you know of his proven worth, that he served with me in the furtherance of the gospel like a child serving his father. Therefore I hope to send him immediately, as soon as I see how things go with me; and I trust in the Lord that I myself also will be coming shortly" (Philippians 2:19-24).
In this passage, Paul is talking about Timothy, not his father. Paul states that Timothy is someone to be trusted and that he has served Paul just like a child serving his father; that is, that Timothy has willingly obeyed Paul. The passage is one of several that shows the close relationship between Paul and Timothy but it tells us nothing about Timothy's actual father.