Is my understanding correct? The position of a "Bible Christian" is that, according to the Bible, only the Apostles themselves were given any kind of "authority" to teach in the name of Jesus Christ.
Authority is in Jesus Christ, not the apostles: Matthew 28:18-20 (NIV) Then Jesus came to them and said, "All authority in heaven and on earth has been given to me. 19 Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, 20 and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age."
Is this authority exclusive to the Apostles, it is not to be handed down, nor is it "transferable" in any way, shape, or form?
Only Jesus' authority is recognized; not apostles, pastors, priests, etc.:
1 Corinthians 3:10-11 (NIV) By the grace God has given me, I [Paul] laid a foundation as an expert builder, and someone else is building on it. But each one should be careful how he builds. 11 For no one can lay any foundation other than the one already laid, which is Jesus Christ.
Ephesians 2:19-22 (NIV) Consequently, you are no longer foreigners and aliens, but fellow citizens with God's people and members of God's household, 20 built on the foundation of the apostles and prophets, with Christ Jesus himself as the chief cornerstone. 21 In him the whole building is joined together and rises to become a holy temple in the Lord. 22 And in him you too are being built together to become a dwelling in which God lives by his Spirit.
2 Timothy 2:19 (NIV) Nevertheless, God's solid foundation stands firm, sealed with this inscription: "The Lord knows those who are his," and, "Everyone who confesses the name of the Lord must turn away from wickedness."
2 Corinthians 5:18-20 (NIV) All this is from God, who reconciled us to himself through Christ and gave us the ministry of reconciliation: 19 that God was reconciling the world to himself in Christ, not counting men's sins against them. And he has committed to us the message of reconciliation. 20 We are therefore Christ's ambassadors, as though God were making his appeal through us. We implore you on Christ's behalf: Be reconciled to God.