Legal Question in Intellectual Property in California

Your question confuses copyright and trademark principles. The short answer, is 'no' you can't use other people's work for your own profit (i.e. paid engagements).

"Fair use" is a copyright principle that permits critics, reviewers and teachers to include small portions of the original in their critiques of a copyrighted work. The use you are proposing sounds as if it is to enable you to profit from someone else's famous work and does not qualify as "fair use" under copyright law. Period. If, however, the work is old enough that it has fallen into public domain, then anyone can use it. The question is whether the original creator properly maintained their copyright, which you would need to investigate. You might be interested to learn that Star Trek itself was the subject of one of the controlling cases in copyright law and is NOT in the public domain.

Generally speaking, logos fall under trademark law as a means of identifying the origin of a product or service (in this case, a television program). Logos belong to the company that originated them and is using them in commerce to market their products and services. And, yes, as you are well aware, Star Trek products are heavily merchandised and the logos are quite valuable. Again, 'no' you can't use them without permission.

Creating your own "approximate version" is still considered stealing someone else's intellectual property and won't protect you. And, please note, photos other people have taken belong to them, not the public domain.

Your best course of action is to approach (or have your sponsors/employers approach) the owners of the intellectual property you want to use and ask them if they would permit you to use it. Sometimes, especially for fan-based activities, the producers/owners of copyrighted programming will allow limited use as a form of free advertising. Or, they might ask for a relatively nominal payment as a gesture of good will. Even though the legwork to track these down may involves a large amount of labor on your part, it is much less time-consuming and expensive than being the subject of an infringement lawsuit. Owners of valuable intellectual property have in recent years become very savvy about protecting it and will take action against thieves. Don't do it.