I think that the current complete ban on third-party/licensed worlds is overzealous.
Research is not worldbuilding
Firstly, a question like "How do lightsabers work in Star Wars?" should remain illegal. Not because it references Star Wars, but because it is not worldbuilding. Asking someone to find the "canonical answer" in a pile of scattered and often contradictory lore is research, and it belongs on SciFi.SE.
An analogous question: "How do Zeus' powers work in Greek mythology?" Such a question would also be forbidden, and it would fit on Mythology.SE instead. A question that asks for a canonical answer, that is answered solely by research, is not doing any worldbuilding.
When questions with third-party worlds are worldbuilding
First for mythology: "Zeus is real, he is a three meter tall alien. [insert complete skillset here]. Could his lightning bolts be produced by an organ like that of the electric eel?" Such a question is worldbuilding, in my book. In fact, it is the biological equivalent of this existing question. The purpose of this question is not to understand a work, but to expand on it. Zeus' myths are many, but even if one of them happens to say just where his powers come from, stating that explanation is not by definition the most valid answer. That is the difference with SciFi.SE, where canonicity is a criterium for answer validity (even if speculation within the bounds of canonical rules is allowed).
An author building a world can use any works they want for inspiration. A fanfiction author is one who is expanding on a setting that already exists. That includes mythology - there is fanfiction for the Bible. The more ambitious fanfiction authors do not just utilise the same characters in canon to tell new stories with, but they expand the world as they do so, to facilitate the story they want to tell. Sometimes doing a lot of worldbuilding is necessary, if the original left a lot unsaid.
And that is where a platform supporting worldbuilders, such as Worldbuilding.SE, comes in handy. If one is making a world based off of a different property, the thing being added is still fresh and new, but working under certain confines. Such confines are no any different from the confines of one's own already-established premise.
I quickly made a question to see what this exotic creature, an acceptable third-party-world-based question, might look like.
Should the Water Tribe from Avatar put waterbenders on ocean-going vessels to make them go faster?
The setting of Avatar: The Last Airbender features several kinds of telekinetic magic, one of which is called waterbending and is practised by some members of the nation called the Water Tribe. The wielders of this magic perform various poses (inspired by martial arts) in order to manipulate water, both tiny drops in the air and massive bodies of water - even occasionally human bodies. Some clips to serve as visual references: 1, 2, 3, 4.
Waterbenders are rare, but if a bunch of them worked together, how fast could they propel a vessel over oceanic distances? The goal of the question is to see how a more militant Water Tribe would best spread out their limited supply of benders; whether putting them on ship-propelling duty would make a difference in speed marked enough to make that a full-time job in the army.
The world is based on East Asia around the 18th century, so take a ship the size and design of a mid-sized Junk (reference). With twenty people aboard it should weigh at least eighty metric tons.
The force of waterbending in the show varies from user to user, as well as from moment to moment (as the plot demands it). For the purposes of this question, I will assume they can exert a force of 200000 Newtons on water within reach (30 meters in any direction, no line of sight required), but no more than 30 Newtons per liter. So with more water they can do more. They can sustain that force for two hours a day, in spurts of fifteen minutes with rests in between.
There are three waterbenders taking turns, and they move the ship by moving the ocean underneath it. How fast could a cruising speed could they maintain over long trips? Would that speed be sufficiently greater than average sailing speed to make this task a worthwhile one?
The precise role of canon
The biggest part of making third-party-world questions compliant with Worldbuilding.SE is to affirm that the "correct" answer is not automatically the best answer. Most of these questions frankly would not have correct answers to begin with because worldbuilding fanfic authors are by definition pushing boundaries. Canonical information can however contribute to an answer, even canonical information not given in the question body. I have thought about this for a while, and I think I can justify that we take the same approach that we do to world history. Any "canonically accurate" information should be treated like "historically accurate" information.
There are no documentary writers on this site so every real-life question is at least for historic fiction, basically a fanfic of world history. That means that if a question is e.g. about what churches would do with orphans in the middle ages, the correct answer (a listing of their historical practices to our best understanding) can be a good one, but so can a bit of intriguing writing that is both believable and makes for a good story.
Similarly, an answer to the Avatar question mentioning waterbenders utilising the full moon, when they are at their strongest, is using "canonical" information outside the question body. The responsibility of the querent is to specifically mention every part of the fictional system that they want an answer to address. The full moon factor is not something to just expect answerers to know, and if the querent corrects people who fail to mention it, then they are essentially moving the goalposts. So they should add enough information to the question body, that they are comfortable with someone who knows nothing else of the source material to answer it. Someone who is familiar with the lore may have even more ideas, but that would just be a bonus.
I suggest a tag third-party-world-based, and if possible I would suggest that said tag adds a warning above the question (like hard-science has), so site visitors understand the same principles I just laid out:
This user is building their world based on a third-party world. The purpose of this question is to expand on the source material, not to merely expand one's understanding of it. Canonicity with all of the original lore is not a criterium for answer validity, only with what has been established in the question body.
The tag description could read:
Use this tag when some of the rules or aesthetics of your setting are explicitly derived from one or more existing settings. Ask only questions to expand on those worlds with worldbuilding of your own. Finding canonical answers is not what Worldbuilding.SE is for; see SciFi.SE or Mythology.SE instead.
One could even argue that this tag should be retroactively applied to all questions based on existing mythology.
About world ownership
Some of elemtilas' thoughts deserve a response not spread out over six comments. Firstly, I am aware of the ethical issue with fanfiction. Strangers playing with their intellectual property is something that some renowned authors absolutely detest (and others welcome), and while lawsuits are unlikely due to the fanfic authors making no money of their unlicensed works, it is a completely fair view to be against the practise altogether. If an ethical reason is one why Worldbuilding.SE cannot support fanfiction, then so be it - I have nothing to say in response.
Another argument however is that of whether the world being owned by someone else would hamper the ability of the querent to judge the best answer. Now again, from a legal point of view a world absolutely has an owner (unless it's public domain), and arguably so from an ethical point of view. But worlds live not just on paper, but also in the heads of those who have read it. If you are passionate enough about a world - which fanfic authors are without exception - then you will be having your own conception in your head of every rule from the source material that you wish to implement in your work of fiction.
A fanfic author chooses whether and in what respects to satisfy the rules of the source material. Both softer thematic rules and concrete systems. And the rules that authors do choose to satisfy to the letter, they will tend to understand how that rule is applied. That means that a querent asking about a fanfic will absolutely be able to judge the best answer.
This is even more obvious for the medium of crossover fanfiction. Someone writing a serious crossover fanfic will need to do their own analysis on the source works, as rigorous as they want the resultant work to be.
Would Darth Vader's sword cut through Captain America's shield? No author, living or dead, can state that with authority. Not even if we put George Lucas and the revived body of Jack Kirby in a room together. But an author who does their own analysis of just how a lightsaber melts things and what sort of things have damaged vibranium over eight decades of canon, and then presents their findings to Worldbuilding.SE and asks whether their conclusion holds water... are they not welcome on the site by definition? I think they should be. They are asking about their version of the sources, their interpretation of it. They started doing worldbuilding the minute they started applying the results of their analysis - which become new rules. That is also exactly what would make them unwelcome on SciFi.SE, where others can have a different interpretation of the source works and it's all fair game.
Why not just use generic water mages?
Mentioning the setting is the equivalent of adding paragraph upon paragraph of flavour text, background and aesthetics that, while not crucial to the question (crucial details must be mentioned in the question body), can serve as inspiration for the creative writers of Worldbuilding.SE to hook on.
Using generics and obfuscating the source material is the current solution for fanfic author on this platform. They are to neglect from naming their setting, to forego providing its many visual references because as soon as one mentions a word that should have a ™ behind it, the close-votes quickly rack up. This has hindered some questions of mine, and possibly some of others too.
For clarification of the current rules. These questions are worldbuilding. There are no ocean-going waterbending-propelled vessels in Avatar; this author is inventing them within some of the confines of the source material. And when those confines are clearly specified, it is no different from giving the premise of one's own setting.
The current rule is just overzealous, banning words instead of banning bad questions. The nuance between asking about worlds and expanding on worlds is a vital distinction, and by overtly making it, the site could not just facilitate fanfiction: it would improve its very definition of worldbuilding.