First of all, the practice of infant baptism has no foundation in Scripture. Second, using "hard cases" such as a person dying right before baptism is grasping at straws. God is just and He has the final say on who is saved. However, we on this earth only have the Bible record to show us what He commands of us in order to receive salvation.
Regarding 1 Corinthians, Paul is speaking to members of the church at Corinth. These people are saved. In the first few verses you quoted, no, he does not specifically talk about baptism. But he does refer to the Gospel that he did preach to them. In Acts 18, we see Paul preaching to those in Corinth. In verse 8, it says "and many of the Corinthians hearing believed, and were baptized". If baptism isn't a part of the Gospel, then how did they know to be baptized?
As you noted, he does talk about baptism later in the same chapter. However, Paul is not saying baptism is not a part of the Gospel. You even included verse 13 which shows he is talking about those in Corinth who were claiming the name of the one who baptized them, not the name of Christ. His problem was not with baptism, but with claiming the name of the baptizer.
Tough to builds a doctrine on the words that are not there huh?