Both are techinically correct, but the version without will is used more often. A "that" could also be inserted as well. "I hope that you have a good time." But, to be honest, it really is just a case of where you want to stress meaning in the sentence.

"I hope you have a good time" is totally acceptable!

Yes, it's fine, and you don't need the 'will'. The more words you add, the less natural it becomes. So, while 'I hope that you will have a good time' is grammatically correct, we probably wouldn't say this. 'I hope you have a good time' is much more natural.