If you look at the context of John 11 you will see what Jesus meant. I will have to say I can see why you found this passage confusing at first. In John 11:17-27 Jesus talks to Mary about the death of her brother Lazarus. He tells her “Your brother will rise again.” (v. 23). Mary understands Jesus to be talking about the final resurrection at the end of time (Daniel 12:2-3), not about physical resurrection. Then Jesus tells her what you quoted above, “Everyone who lives and believes in Me will never die.” (v. 26) What does he mean? Does he mean that they will not die physically? Obviously not!!! We already know that Mary was thinking about the final resurrection, and that is what Jesus is talking about. Jesus had just told his own followers that he would die physically. Jesus was well aware that the fate of all humans before his return is that their bodies die and decay. Obviously, Jesus was not talking about that. He was talking about the final resurrection, after which will come judgment and, for those who believe in Jesus, everlasting life. This is talked about in Daniel 12:2-3, Matthew 25:46 and elsewhere, Revelation 20:11-15 and elsewhere.
The words “darkness” and “thirst” are used metaphorically here. Jesus used a lot of metaphors to describe his teaching, especially as recorded in John. Jesus said, “I am the door.” He is the spiritual door. Jesus said “I am the way.” He is the way to God. He said the he is the light. Here he is talking about spiritual, not physical light, and when he talks about darkness (for example in John 3) he is talking about living a life outside the knowledge of God. When tells us that we will never go thirsty to the woman at the well, he is talking about spiritual thirst. In that passage Jesus told the woman that his food was to do the will of God. Obviously, all of these uses are metaphorical, as both his friends and enemies recognized.
Why the Pharisees rejected Jesus is hard for me to say for sure. You would have to ask them. Unfortunately, you cannot. We have some hints in the New Testament. For example in John 11:47-48, right after Jesus raised Lazarus, the Pharisees met with the Sanhedrin to plot to kill Jesus. Why? If they do not stop Jesus, the Romans would take away their power to govern! It was sheer greed and desire for power that caused them to not believe. In this case, did they believe that Jesus raised Lazarus from the dead? Absolutely!!! That is why they wanted to kill him. Did they know that he fulfilled the prophecy to come into Jerusalem on a donkey (Zechariah 9), to be betrayed for thirty pieces of silver (Zechariah 11), to be crucified (Psalm 22)? Yes they did!!! Yet, they refused to believe that he fulfilled the Prophets. How did this work? Why did they not believe the evidence right in front of their eyes? I suppose it was a variety of reasons, but according to John 11, one of those reasons was jealousy. They did not want the people to follow Jesus. They wanted the people to continue following them. They also wanted to protect their power and wealth. Was that the reason in every case? Perhaps not. Perhaps there were other sins people wanted to hold onto, or even other unknown reasons. I do not know. However, here is what I know. It is not for lack of evidence that Jesus fulfilled the prophecies.
Why did they tell the guards to tell people his body was stolen? For the same reasons. They hated the truth. They denied the truth, because they would lose their position of wealth and power, so they covered it up. What is else is new? Is this not what politicians and people in power have done throughout human history? (not all, of course, but the majority)