True and the following are the traditions in which Paul is teaching Timothy to follow:
The first being what follows exactly after the above statement:
2 Timothy 3:
14 But continue thou in the things which thou hast learned and hast been assured of, knowing of whom thou hast learned them; 15 And that from a child thou hast known the holy scriptures, which are able to make thee wise unto salvation through faith which is in Christ Jesus.16 All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness: 17 That the man of God may be perfect, thoroughly furnished unto all good works.
Right there, Paul clearly points out what he teaches Timothy can be verified with Scriptures.
The second being what Christ taught:
Luke 24:
44 And he said unto them,These are the words which I spake unto you, while I was yet with you, that all things must be fulfilled, which were written in the law of Moses, and in the prophets, and in the psalms, concerning me. 45 Then opened he their understanding, that they might understand the scriptures,46 And said unto them, Thus it is written, and thus it behooved Christ to suffer, and to rise from the dead the third day: 47 And that repentance and remission of sins should be preached in his name among all nations, beginning at Jerusalem. 48 And ye are witnesses of these things.
The above is the tradition passed from Jesus Christ. It was also the Gospel preached by all the apostles and disciples of Christ.
Confirmed yet here as well:
John 20:
30 And many other signs truly did Jesus in the presence of his disciples, which are not written in this book: 31 But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name.
How was this applied in the ministry of all the apostles? The apostle Paul makes it clear:
1 Corinthians 15:
15 Moreover, brethren, I declare to you the gospel which I preached to you, which also you received and in which you stand, 2 by which also you are saved, if you hold fast that word which I preached to you—unless you believed in vain.
3 For I delivered to you first of all that which I also received: that Christ died for our sins according to the Scriptures, 4 and that He was buried, and that He rose again the third day according to the Scriptures, 5 and that He was seen by Cephas, then by the twelve. 6 After that He was seen by over five hundred brethren at once, of whom the greater part remain to the present, but some have fallen asleep. 7 After that He was seen by James, then by all the apostles. 8 Then last of all He was seen by me also, as by one born out of due time.
9 For I am the least of the apostles, who am not worthy to be called an apostle, because I persecuted the church of God. 10 But by the grace of God I am what I am, and His grace toward me was not in vain; but I labored more abundantly than they all, yet not I, but the grace of God which was with me. 11 Therefore, whether it was I or they, so we preach and so you believed.
So we see exactly what is meant by tradition or a term you are more familiar with, what is Apostolic Tradition. In the NT this is clear, it is the declaration of Jesus Christ as the Son of God, and Gospel of Christ crucified, died, and Resurrected; Ascended into Heaven to be seated at the Right Hand of the Father; and He will Come Again. This is the tradition passed down, written down and became the Rule of Faith in the post apostolic era. We see this here:
. . . this faith: in one God, the Father Almighty, who made the heaven and the earth and the seas and all the things that are in them; and in one Christ Jesus, the Son of God, who was made flesh for our salvation; and in the Holy Spirit, who made known through the prophets the plan of salvation, and the coming, and the birth from a virgin, and the passion, and the resurrection from the dead, and the bodily ascension into heaven of the beloved Christ Jesus, our Lord, and his future appearing from heaven in the glory of the Father to sum up all things and to raise anew all flesh of the whole human race . . . St. Irenaeus of Lyons ( The "Rule of Faith" as recorded by Irenaeus) http://www.creeds.net/ancient/Irenaeus.htm
Irenaeus expands this understanding in Against Heresies Book 3.2.1:
1. When, however, they are confuted from the Scriptures, they turn round and accuse these same Scriptures, as if they were not correct, nor of authority, and [assert] that they are ambiguous, and that the truth cannot be extracted from them by those who are ignorant of tradition. For [they allege] that the truth was not delivered by means of written documents, but vivâ voce: wherefore also Paul declared, “But we speak wisdom among those that are perfect, but not the wisdom of this world.” 1 Corinthians 2:6 And this wisdom each one of them alleges to be the fiction of his own inventing, forsooth; so that, according to their idea, the truth properly resides at one time in Valentinus, at another in Marcion, at another in Cerinthus, then afterwards in Basilides, or has even been indifferently in any other opponent, who could speak nothing pertaining to salvation. For every one of these men, being altogether of a perverse disposition, depraving the system of truth, is not ashamed to preach himself. http://www.newadvent.org/fathers/0103302.htm
Where?
The above is the tradition passed from Jesus Christ
Yes. Nowhere does it say that everything He passed to us was written down; in fact, John 20:30 that you cite, and John 21:25 say the opposite.
Neither of the two quotes from St. Irenaeus say that the Apostolic Tradition is entirely contained in the Scripture. The first quote says what it is, not where it is written; and the second says that one should not contradict the Scripture. Both are Catholic teachings.