To be honest, your writing sounds rather like you have been convinced of their doctrine. I'd encourage to you to look at their claims more critically and compare it to Scripture.
Regarding the so-called passage about Peter (we're all saints in Christ, so it's silly to call him Saint Peter anymore than I'd call you Saint Angel), let's take a look at the passage in context.
"When Jesus came into the coasts of Caesarea Philippi, he asked his disciples, saying, Whom do men say that I the Son of man am? And they said, Some say that thou art John the Baptist: some, Elias; and others, Jeremias, or one of the prophets. He saith unto them, But whom say ye that I am? And Simon Peter answered and said, Thou art the Christ, the Son of the living God. And Jesus answered and said unto him, Blessed art thou, Simon Barjona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven. And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it." Matthew 16:13-18
The context shows that Jesus had asked His disciples a question, and Peter had answered it with a statement of faith - "Thou art the Christ, the Son of the Living God." This is the 'it' that Jesus later told him had been revealed to Peter by the Father. He follows that up with a play on words. You see, 'Peter' in Greek is 'Petros'. The word Christ used for 'rock' is 'petra'. 'Petros' refers to a pebble; a small stone, while 'petra' refers to a large rock; bedrock. Peter himself was only a pebble, but the statement of faith he spoke was to be the bedrock of Christ's church - the fact that Jesus is the Christ (Messiah), and Son of God. You see? Jesus wasn't establishing Peter as the rock of the church (unstable Peter? that would be a poor foundation indeed). Rather, it was the illustrated faith and confidence in Jesus the Son of God that would be the bedrock of all believers.
Furthermore, the Gentile Church (which is what the first church established in Rome would be primarily comprised of) was largely evangelized and led by Paul. Peter stayed in Jerusalem and was a chief elder to the Jewish believers.
As far as Mary, she is called a co-mediatrix by the Catholic organization. This means they believe that she is a co-mediator between God and man. This is blasphemy. 1 Timothy 2:5 says: "For there is one God, and one mediator between God and men, the man Christ Jesus." Also, Hebrews 1:2-3 says: "...his Son, whom he hath appointed heir of all things, by whom also he made the worlds; Who being the brightness of his glory, and the express image of his person, and upholding all things by the word of his power, when he had by himself purged our sins, sat down on the right hand of the Majesty on high:"
The co-mediatrix doctrine is also where they get the idea of Mary praying for us. But as we see from 1 Timothy, the only mediator between God and man is Jesus - not Mary, not the saints. Only Jesus.
As far as that passage in John, let's take a look at the wording:
"When Jesus therefore saw his mother, and the disciple standing by, whom he loved, he saith unto his mother, Woman, behold thy son! Then saith he to the disciple, Behold thy mother! And from that hour that disciple took her unto his own home." John 19:26-27
First, a grammar lesson. The translators of the KJV were attempting to translate languages that used both singular and plural second-person pronouns (unlike modern English, which uses the generic you/your/yours for both singular and plural uses). In order to translate the sense of the words accurately, they chose to use thy/thine to translate singular pronouns, and you/your for plural pronouns. For this reference to be applicable to all of us, Jesus would have had to say 'behold your mother' (as you incorrectly quoted). Rather, he said 'behold thy mother'. The use of the word 'thy' indicates that he was speaking to a singular person - the disciple He loved (i.e. John). The Catholic understanding that she was given to all of us is thus grammatically inaccurate.
The Catholic organization also teaches that Mary was herself immaculately conceived (i.e. of a virgin) and without sin. This extrabiblical teaching would make her the promised Messiah, not Jesus - you see the doctrinal problem with this? They call her 'mother of God' - but God by definition has no parent - only the human body of Christ had a mother. They also call her Queen of Heaven. The Bible references a Queen of Heaven. She was a false goddess sinfully worshipped by the Israelites in the time of Jeremiah (Jeremiah chapters 7, 44). The Catholic representation of Mary thus traces back to an ancient false religion that provoked God's wrath against the Israelites.