it stands to reason that you include the other iconic things as well. Klingons, Scotty being drunk, "I'm a doctor, not a ____", etc.
God no. I'd rather see less mistaking of reference
in general, and to leave grinding out the old cliches to sitcoms and parodies.
As for whether it should be something outright new: it would only be genuinely a "reboot" if it was a ground-up fresh run at the Great Bird's original set of ideas. That original set of ideas is actually quite basic and can be taken a lot of very different directions, but I'd say if you're using them as a core it's appropriate (and more than just cynically cashing in on an established brand) to call it Trek.