It's really just a matter of personal preferences. Some people like reading about canon characters others (like me) rather read about original characters. However fan fiction is best approached as a way to express your own interests instead of catering to your audiences as your chances for professional publication are slim to none.
See above. I believe the 50 Shades of Gray writer changed her fan fiction (whatever it was? Harry Potter? Twilight?) into an original novel which then got published. Trying to get fan fiction published is almost impossible. But you would most likely have to write about canon characters and submit your work (via an agent) to the publisher who is most likely not accepting any unsolicited material.
Up to you when you set your story. Again don't try to follow trends, write what appeals to yourself. Stardates and other established Star Trek tropes can help to make your story more authentic. No point writing a Trek story if it doesn't feel like one.
How authentic do you want to be? A lot of technobabble is made up. But something sounding authentic or actually being scientifically accurate is more likely to impress readers (if they understand it). Obviously made-up stuff is likely to be a turn-off as readers may not take your work seriously or feel that you haven't.
If you want to write fan fiction because that's what you think you enjoy doing, knock yourself out and writer whatever you think you'd enjoy reading. If it's good, people will read it. If you're looking to be a published author, use fan fiction as a way to hone your skills but start thinking about working on original stuff.
Also, do a lot of reading. In my opinion, too many fanfic writers figure that they are awesomely talented without ever reading anything else. Truth is most writers aren't good at writing until they're well read.