No.

Incorrect.

They base their religion on hate for other peoples religion.

Incorrect.

If they are truly Satanists,

The Mission Of The Satanic Temple Is To Encourage Benevolence And Empathy, Reject Tyrannical Authority, Advocate Practical Common Sense, Oppose Injustice, And Undertake Noble Pursuits.

then they know for a fact that they oppose God, and are in the wrong.

If you haven't read the bible you might get that impression. A satan is an adversary in Hebrew. There are passages (Numbers 22 for the earliest one by date of the text) where the word refers to god or the angel of god who is operating as an adversary.

This is not a hard concept to understand.

And yet, here you are.

This is what defines them as a hate group.

No, they don't teach hate. They encourage benevolence and empathy.

It is in fact Atheists, and atheist ideals that define the separation of Church and state.

It's American ideals. But it certainly helped that the founding fathers were secular thinkers.