Because without understanding it at least a little bit, you can't understand classical literature at all. Without classical literature, it's hard to see how our own literature developed.
Opinions on this vary. But those who don't want to have religious instruction in school are probably afraid that a certain religion will be more or less imposed on children. After all, if a religion is to be taught, who should decide which religion will be taught? Would you agree with your children being taught (for example) Buddhism if you are a Christian; or Christianity if you are a Buddhist, because somebody else decided that that is the "best" religion? Those who do want religion may also have valid arguments - for example, they might consider that religion to be the source of morality/ethics. However, the problem persists, which religion should be taught. Of course it might be possible to teach several religions, depending on the children's preferences (or those of their parents). But that will increase the cost for the school.
euripedes was a philosopher that taught happiness
No, it is taught to you.
Catholic religion and Greek Christianity are somewhat similar, but other Greek religions are not similar to the Catholic religion.
In ancient Greece, the works of Homer were so highly regarded as examples of Greek literature, that they were universally the textbooks of choice for learning to read and write Greek. They were also, in many ways, the 'Bible' of the Greek religion, so that students were learning the fundamentals of both Greek and religion. Even as late as the nineteenth century, some parents taught their children to read from the Christian Bible, if formal schooling was impracticable.
Well, slightly less history... But to be serious, they were taught Greek, religion (i.e. the Protestant religion usually) and mathematics. Philosophy, Literature (for example Chaucer. Greek playwrights, etc), as well as rhetoric (speaking out loud) were also studied.
I think religion should be taught in public schools. It should not say this is what's right to believe and this is what's wrong, but students should be educated about the most predominant religions. It would allow students to get a better understanding of something and allow for more acceptance and maybe some will find a religion that suits them more.
what taught about relationships(Parent-Child)
No, not really. The only 'religion' they have is greek!
musilms
Muhammed (pbuh) taught his religion which was Islam.
Cleopatra was Greek, so she followed the Greek polytheistic religion