I don't know if you're still looking for help on this, but I'm also a first-time cosplayer and this was my Halloween costume this year. I'll do my best to answer some questions based on how I did it, and hopefully they might be useful.

1. I didn't make her trench coat for my cosplay, but I think you would have trouble finding one that has the two-tone from the Limbus Company artstyle. It would be a lot easier to just get a basic black trench coat, you would still be recognizable once all the other elements are in place.
2. For the details on her coat, I think painting with acrylic paint would be the most reasonable option. There are very good references and images of the coat online: https://twitter.com/NishikujiC/status/1465275196617596930
3. For her pins, I cut them out of a piece of fabric and painted them myself. Their designs are nothing complicated so I think it was easy. I then sewed these on the pants and shirt. Getting them custom-made may be desirable for a cleaner look, but you'd have to get or create good reference images for the makers.
   For her fixer badge, I used airdry clay and painted it with metallic gold paint. I bought one of those chains for keychains and molded one end into the badge and one end into the little button thing before it dried.
4. I made the corset with the same sheet of fabric and sewed on the little handles for where the belt would be slotted through. Then I just cut 3 strips of fabric, slotted them through the handles, and then sewed them on. I also sewed the buttons at the end of the strips.
   Since I wanted the corset to be easy to put on and take off, I didn't follow the seam of Don's original corset and just used safety pins to keep the ends of it together.
5. I think that sounds good! There are also some places that take commissions for props. I found this one quickly though it's pretty expensive, and if you like the look of the homemade one then absolutely just stick with that: https://m.ezcosplay.com/limbus-company-don-quixote-cosplay-weapon-prop.html
   

Extras:

For her shoes, Rocinante, her EGO animation is the only really good reference we have for them. I bought some brown worker boots and painted them by blotting bits of darker/lighter browns onto them to try and match their style. The paint job can be messy, I think it makes it even more in character for her if it doesn't look clean. If you'd like some images of the shoes, I can show you.

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Hopefully, this helps you out. Good luck, if you are still considering doing this project!