What is the Biblical basis for the special mediatorial position of the Pope, Cardinals, etc in the Catholic church? Following-up a previous question, what Biblical basis does the Catholic church claim for the special role of Priests, Cardinals, and Popes in the structure of the church?
The Pope

The Pope is the Vicar of Christ on earth, the direct successor of Peter who was handed the Keys to the Kingdom by Jesus. (Matthew 16:19)  This makes his role a mediator for consciences, or an arbiter of what is right and wrong, true and false in terms of faith and morals, but not science and logic. (ccc 891)

Bishops

Catholics claim apostolic succession, meaning that what Jesus said the His apostles applies specifically to the episcopate, the Bishops especially. 

So, when Jesus says:


  Then Jesus approached and said to them, “All power in heaven and on earth has been given to me. Go, therefore, and make disciples of all nations, baptizing them in the name of the Father, and of the Son, and of the holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, until the end of the age.” 


(Matt 28:18-20 NAB)

He is commissioning the Apostles and their successors to act in His name.  Not baptizing like John baptized, but baptizing like Christ baptized.  

Jesus established or sanctified all 7 sacraments in his short ministry and expects His priests to carry them out. 


Baptism (John 3:1-5)
Confirmation (Acts 8:5-17 and Acts 1)
The Eucharist (John 6:48-58, and institution narratives in synoptics) 
Holy Orders (Everything Jesus tells the apostles to do)
Marriage (Mark 10:1-12, Eph 5)
Reconciliation (John 20:19-23)
Anointing of the Sick (Mark 6:7, 12-13)


Biblical Sources for sacraments

Priests

Priests (as well as the Pope and the Bishops) act 'in persona Christi'.  They forgive sins ,through God, because Jesus forgave sins and told the apostles to act accordingly (John 20:23).  The very definition of priests is one who offers sacrifices for the people.  You don't have to poke very far into Leviticus to find a biblical basis for priests acting like priests.  Priests in the new covenant 'do this in remembrance of me', and they say this, not 'do this in remembrance of Jesus' during the Holy Sacrifice of the Mass in the Eucharistic prayer before communion. 

Cardinals

One thing to point out is, not all cardinals are currently Bishops (and historically there have been lay cardinals, so it's an error to conflate their position with Bishops, they do not act as cardinals as mediators in any way.