Regarding I Timothy 1:4-6: In verse 4 what are the myths and endless genealogies being talked about and in verse 6 and verse 3 who are the some men or the certain men that are teaching false doctrine?
"As I urged you when I went into Macedonia -- remain in Ephesus that you may charge some that they teach no other doctrine, nor give heed to fables and endless genealogies, which cause disputes rather than godly edification which is in faith. Now the purpose of the commandment is love from a pure heart, from a good conscience, and from sincere faith, from which some, having strayed, have turned aside to idle talk, desiring to be teachers of the law, understanding neither what they say nor the things which they affirm. But we know that the law is good if one uses it lawfully" (I Timothy 1:3-8).
Paul is referring to same time frame that is recorded in Acts 20:1.
The doctrine Paul is concerned about maintaining is that of the Gospel. "If anyone teaches otherwise and does not consent to wholesome words, even the words of our Lord Jesus Christ, and to the doctrine which accords with godliness, he is proud, knowing nothing, but is obsessed with disputes and arguments over words, from which come envy, strife, reviling, evil suspicions, useless wranglings of men of corrupt minds and destitute of the truth, who suppose that godliness is a means of gain. From such withdraw yourself" (I Timothy 6:3-5).
We know from history that apocrypha was a popular style of writing during this era. People would create documents that imitate the sacred writings and often used pen names of famous biblical characters to give a feel of authenticity to their writings. Both the Jews and Gentiles collected teachings and argued at length over their meanings. However, the problem is that such treated all works equally and the teachings of the Scriptures are not on the same plain as fables. "O Timothy! Guard what was committed to your trust, avoiding the profane and idle babblings and contradictions of what is falsely called knowledge -- by professing it some have strayed concerning the faith" (I Timothy 6:20-21). The Talmud of the Jews is a collection of such stories. That Paul is probably more focused on fables of Jewish origin can be seen in: "This testimony is true. Therefore rebuke them sharply, that they may be sound in the faith, not giving heed to Jewish fables and commandments of men who turn from the truth" (Titus 1:13-14).
The Jews were heavily into ancestry since they saw themselves as superior because of their lineage. "Therefore bear fruits worthy of repentance, and do not begin to say to yourselves, 'We have Abraham as our father.' For I say to you that God is able to raise up children to Abraham from these stones" (Luke 3:8). The apocrypha writings were filled with such genealogies, supposed that these gave more credibility to the things they wrote. "But avoid foolish disputes, genealogies, contentions, and strivings about the law; for they are unprofitable and useless" (Titus 3:9).
False teaching has always been a problem because it offers people what they want to hear and not what they need to hear. "Preach the word! Be ready in season and out of season. Convince, rebuke, exhort, with all longsuffering and teaching. For the time will come when they will not endure sound doctrine, but according to their own desires, because they have itching ears, they will heap up for themselves teachers; and they will turn their ears away from the truth, and be turned aside to fables" (II Timothy 4:2-4). Such continues to be a problem.
"I marvel that you are turning away so soon from Him who called you in the grace of Christ, to a different gospel, which is not another; but there are some who trouble you and want to pervert the gospel of Christ. But even if we, or an angel from heaven, preach any other gospel to you than what we have preached to you, let him be accursed. As we have said before, so now I say again, if anyone preaches any other gospel to you than what you have received, let him be accursed. For do I now persuade men, or God? Or do I seek to please men? For if I still pleased men, I would not be a bondservant of Christ" (Galatians 1:6-10).