After watching many episodes of many Star Trek series I was considering to make my own fanfic like I did with Half-Life Ampere.

I have a few ideas, but I'm not sure, it is normal to want to make up my own story set in Star Trek universe?

Also, I personally have troubles to try to be the most lore friendly possible and that make me angry while I'm typing down eveything.

There is no "normal" in fanfic other than "I want to write this". I mean, your screenshot has explicit angel/demon sex, with penises, in space.

So: do you want to write a story set in the Star Trek universe? Then write a story set in the Star Trek universe.

In my case I was thinking to make my own Lower Decks or TNG to say something.

And what are you waiting for now?

I'm writing another book right now. I'm waiting to finish other stuff first to not procrastinate.

it is normal to want to make up my own story set in Star Trek universe

with star trek being one of the first really big fan fiction spawning franchises: yes

If you want to do it, just go ahead. It's literally how fan fiction works. Fandom communities aren't clearance offices that can deny or grant you a license to write what you want 👍

Chef Janeway: Coffee and peanut butter and jelly sandwiches.

Chef Picard: Wine. Lots of wine.

Star trek is one of the first TV series to get a serious fanfic community. It's just that that happened in the zine era.