What cutting-edge technology would I like to see in the new show?
Robots.
Not the *special snowflake* Data-kind, but the I, robot-everyday kind. Maybe just show them on Earth and the colonies and explain humanoid robots aren't allowed on starships, because .... ? They got hacked in the past? At least something, that explains why they have never shown up in past series.
Rovers.
You know, the wheel is an invention that's thousands of years old, and still working just fine? I would like to see them cruisin' exoplanets like the guys in Prometheus or the Robinsons in Lost in Space did.
Classic communicators.
Okay, modern mobile phones are already smaller than Trek's communicators. But you knowh what? They are robust military technology, communicating between planets without infrastructure like telemasts. Have you ever seen modern military radios? They're freakin' huge! Also, the tricorders and communicators could be merged, like you said before.
More and smaller medicine technology
I say: Let the doctor wear a small case on his away missions. And in this case are multiple medicine- instruments, with which he can operate in sterile conditions, grow new flesh on the fly, etc. etc. etc.
James Bond gadgets
James Bond has a watch, that can shoot lasers, hack computers and detonate. Also, it has a grappling hook, that can be shot at the ceiling and than pull Bond out of danger. I want to see something like that in Star Trek.
More bugs
Seriously. In Star Trek technology always works and is always compatible. In reality, technology is getting more and more complicated, and is starting to develop more and more a life on it's own. (really, how often have you to either restart your computer or wait until he fixes a problem on it's own, because you yourself aren't capable anymore of identifying and fixing the problems of your computer?)
Genetic meddling
Yeah, that's augments, and it's illegal in the Trek universe. I want to see bad guys using it. Giving themselves genetic improvements, and also augmenting their appereance and their race to hide undercover.
Giant computers running whole civilisations
You know, classic, campy 60s' Trek? Seems more relevant today than ever.
Technology being invisible
I think, the more advanced technology is, the more it tries to blend into the background. It's only on Federation starships where we see the relative robust, but clunky technology. I think on earth, a normal room might look like a room out of the 19th century: Book shelves. A fire place. A couch facing a table, instead of facing to the television (because there is none). And whenever you do need technology, for example for a phone call, you can sit on your couch, and a holographic window will appear in front of you, in the middle of the room, wherever you need it (and might even move with you, when you're walking the apartement). Hungry? Prder some food, and it will get replicated in the middle of your table, and disappear once you have finished. Go outside? Small robots star cleaning the place, pour the plants, and disappear into the wall once you come back.
Working public transportation systems
Something really outlandish and futuristic. Also: More sidewalks and people on bikes. Once you can literally go anywhere in no time (beaming), people might appreciate good old human exercise a bit more.
In general:
I think in the development of "more sophisticated but tinier" technology, we are right now in a phase of fast development. But we can already see the limits, and there absolutely will be an end to how miniaturized technology will get. The focus should lie more on the social implications of new technology, than the technology itself.
As for "which tecnology is absolutely unimaginable now, but is already predicted by Star Trek"?
How about FTL travel? Engines that can take you into earth orbit in minutes while being reusable? Replicators? (As in super advanced 3D printers) Holodecks? (But used for virtual reality, not "trapped in a Sherlock Holmes story"). Try to dive a bit more into that (Can we "replicate" living beings? What will be the usages for a technology that can simulate literally any reality? Will people have long-distance relationships through the holodeck?) What about universal translators? (How much are we actually talking to another (non-)human being, and how much are we talking with an A.I. that imprints it's own manners onto the conversation?)
Also: drones. Imagine a robot walking on a planet and talking to it's natives, but in reality a guy is in a spacehip, sitting on the holodeck, while all the recorded environment by the robot of said planet is projected as a hologram around the human?
One thing that could distinguish Trek from grim dark shows...would be Trek being fun!. 
