I make crochet dolls and patterns for these dolls. If I make these patterns available for free, I believe it counts as non-commercial use. Where can I find out which companies' designs I can use? Google hasn't brought me any good answers yet. I am particularly interested in whether Disney/Marvel, Nintendo and Studio Ghibli accept this.
You can freely make any patterns of your own invention: I assume you are asking if you can distribute patterns that are based on other people's products, such as Disney. For example one could crochet potholders with the image of a Little Einstein character. This is generally not possible without the permission of the design's creator (or whoever owns the copyright). There is a widespread belief that non-commercial exploitation is allowed, under a "fair use" analysis: this is only true to a miniscule degree. If you copy without permission, you may be sued, and your attorney could try to raise a fair use defense. One of the factors allowing exploitation is the commercial nature of the use (where a non-commercial exploitation would be more consistent with "fair use"). Another factor is the "transformative" nature of an exploitation, meaning that the more you change the original, the better off you are. Probably working against you would be "effect on market", since offering something for free is likely to negatively affect the rights-holder's market share. Even if a rights holder does not currently sell a particular version of a product, such as a PDF version of a book, they might potentially wish to exploit a market. Even if there is no current Little Einstein crocheted potholder, your product would clearly have a huge impact on that potential market. Also working against you would be the amount and substantiality of the copying. In text copying, you can usually copy a short paragraph from a book, whereas the designs you are likely to put out there would be pretty much the whole thing.
Counting on a fair use defense is pretty risky, unless it is based on one of the two fairly clearly established types of use (text copying of short illustrative quotes to prove / exemplify a point, and private-use copying of 40 pages or less of a publication). As far as I know, Disney is very protective of their property rights. The best way to determine if a particular use is allowed is to describe the use and formally request permission from the rights-holder. If they say "No, absolutely not", that would be a reason to ask your lawyer if s/he thought that you could mount a fair use defense against infringement.
Another issue to consider is whether the design enjoys protection as a trademark. Unlike copyright, trademark is not automatic, but you can search the US trademark database, though actually finding things is not trivial. In US law, there are two types of fair use defenses for trademark infringement. One is "descriptive" use, which address the fact that Apple computer company has a registered trademark on "Apple", but that doesn't mean that grocery stores have to start calling the tree fruit "pomme". Conceivably, some company might register a drawing of a quarter moon as a trademark, but that does not mean that nobody can then use a quarter moon as a graphic element. When someone uses an existing image / design as a trademark, they have much less control over use of that design. Little Einstein Leo would be totally outside the realm of descriptive fair use.
The other version is "nominative" fair use, where you are referring to the product, using only as much of the trademark as is needed to identify the thing, and not implying endorsement by the trademark owner. I could refer to Boeing, the airplane maker, as "that plane maker with facilities in Renton", or I could use their trademark. But using their circle-swoop-plane graphic would be beyond nominative fair use. What you are describing is well beyond any fair use concepts for trademarks in the US. Unlike copyright where some version of "fair use" is universal, trademark fair use is not universally recognized.