In my experience, "wish" is no longer used commonly as it is used in your first two sentences.
Merriam-Webster gives this as one of its definitions:
to request in the form of a wish : order
The first sentence is clearly expressing an order. It is not so much "I want you to find it" as "Find it". This use of wish is in my opinion rather rare, although I would not look twice if I found it in Shakespeare. Today we would more likely simply use a command form.
**For clarification on that: If anyone said "I wish that you find me a magic sword", I would consider it wrong, because we do not use "wish" as a command in modern speech.
I would see the second sentence as problematic simply because it omits "would" (or "could"). Typically, if one wants to express a desire or wish with the verb "wish", as your second sentence, it is written like so:
I wish that he would bring me a new book.
I wish I could do it over.
(In a counterfactual clause, one uses the past subjunctive, which is the same as the past tense except for to be: "I wish I were Steve Jobs.")
I'm not sure about the grammatical accuracy of "I wish he bring me a new book", but regardless, it sounds off and I wouldn't recommend it.
The difference between 1) and 3) is that in 1), wish is used not as an expression of hope, but as a command, as the dictionary definition I previously presented.
These, however, are approximately the same, when "wish" is used with "would" to express hope:
I hope [that] you find me a magic sword.
I wish [that] you would find me a magic sword.
Depending on context, "hope" may express more "realistic-ness" than "wish", when used to express a desire or wish. For example:
I hope [that] I get the job at McDonalds.
I wish [that] they would make me CEO of all the food chains in the world.
"that" is omittable in all these sentences.