1. Baptism is not a requirement for salvation. Just take a look at the man on the cross next to the Lord Jesus. The Lord said to him, today you will be with me in paradise. He was never baptised.
For one, you don't know he was never baptized. Two, he was under the old covenant. Three, Jesus' command recorded in Matthew and Mark, that disciples be baptized, came after this incident. He didn't provide exceptions to this command. Your injection of an exception is not found anywhere in the text.
Some people may quote verses which they think prove otherwise but the Bible does not contradict itself, it is simply they don't fully understand the verses they are quoting.
This passage doesn't contradict those passages. Correct. It was a command.
Furthermore, to claim baptism is required for salvation means you are also relying on works to save you. Salvation is by faith alone, not by anything you do (works). Eph 2:8-9
Works do save us. The works that Paul excluded were distinct and specific. In Ephesians he is excluding man-invented works of merit, not all works. He refers to baptism in verses 1-6 and tells them that this work is what saved them and that it was the grace of God. When he gets to 8-9 he tells them that the work they did do was "not of yourselves." Then he goes on to say in verse 10 that we were created to do God's works. Baptism isn't a work invented by man, it's a work invented by God and made effective through Christ's sacrifice.
"The baptism of John, did it come from man or from God?"
In Romans, Paul excluded the works of a then obsolete law. At the time that law was in place, those under it could only be saved doing its works. Once the new covenant was established, you could no longer go back to it and be justified.
"This is the work of God, that you believe in the one he sent."
If we aren't saved by works, then we can't be saved at all. Belief, Jesus says is itself a work.
Remission of sins comes through baptism. All you have to do is read the book of Acts and you see that the apostles never ceased to obey Jesus' command to baptize believers. Why? To wash away their sins. That's what it's for.