Now for the dark side: When you know your art isn't going to get approved but you want to sell it anyway.
I am not saying anyone should do this or that it's a good idea or that it's a respectable thing to do. It is very much in the gray area of what is "allowable". Print-on-demand sites let a lot of stuff slide because if you aren't "trading on the brand's name", the brand isn't likely to notice or, in some cases even care.
It's very simple, but not without risk. Proceed with caution (and for godsake back up your artwork!)
Don't tag your art with anything the brand owns or looks for.
Don't use the brand's name, the actor's names, anything. This will make it harder to find. My Garak stuff is largely invisible because there is no "DS9", Star Trek, etc etc.
So how will people find it? They won't - unless they know to look for it. You can use fandom tags like Garashir, Spirk, Quodo (don't judge me), and the people who want it will be able to find it.
WARNING: your stuff can still get pulled down. And if there's enough of it pulled down, they could shut down your store. So proceed at your own risk. If you have non-fandom art in different subject areas, keep them separate with a different profile and email account. That will minimize, but not eliminate, the likelihood of collateral damage if your fandom art store gets shut down.
Also, know your brand and show runner. Hannibal artists get free reign because Bryan Fuller supports fan artists and fan fiction writers. Same with Neil Gaiman and Good Omens. CBS however will roll over your art like a lawnmower if you tag it #StarTrek. Every brand is different and you have to decide if you want to take the risk.
I have art on both side of the line because I have clients on both side of the line. At any point CBS can decide that not tagging my work isn't good enough and have all my stuff removed, or ask that my shop be shut down. It's worth the risk to me but it might not be for you. Tread carefully and let me just say it again for those in the cheap seats - Back Up Your Art!