hope and the future tense?

Subjunctive voice would impose present tense in the second clause.  Hope typically invokes subjunctive voice.  the hope is from now into the future, in itself, basically, so whatever you hope now is spoken of as now.

You can, in some cases, put the secondary clause into the future, if you want to declare that it is only some not-near future that is the object of your hopes.

Like, I hope that you will some day find a good wife.  Contrast with "I hope the weather gets better".

In your case, I wish you a happy new year, I hope you have a good new year, but I hope the new year will be good for you.

Generally, the statement is "Happy New Year." If you're saying it, there is no need for "I hope" and "have a" is understood.

Both are okay.

"I hope you have a happy New Year" is more common.  In this case, the future tense is understood because of the fact that it starts with "I hope."  The word "will" which normally expresses future tense is omitted but implied.