Yes, but that's because the name Peter Pan, as it refers to the name of the main character in the story that is now in public domain, was not under any trademark in the first place (any trademark status it has since acquired, such as the name of a particular brand of peanut butter, or bus services, or the like, happened after the work fell into public domain).
First, names are not copyrightable; even during the time when the original Peter Pan plays by Barrie were copyrighted, the name could have been used by others without running afoul of the copyright.
Second, while patents require novelty, i.e. that something has never been done before, trademarks do not. And while copyrights require originality, i.e. that something originates from an author, rather than having been copied, trademarks do not. It is perfectly acceptable to use pre-existing material to make a mark, e.g. APPLE for computers, and it is even perfectly acceptable to copy someone else's mark provided that it is not then used in an infringing or diluting manner. A classic example is the PERSON'S mark used for clothing: In 1977, a Japanese clothing firm began using the PERSON'S mark in Japan, and in 1981 an American visited Japan, bought some samples of PERSON'S clothing, took them to America, and began manufacturing and selling his own line of clothing in America using the PERSON'S mark. In 1985, the Japanese firm began selling in the US, and the next year, the two firms realized they were now both in the same market with the same mark. They went to court, and ultimately the American won, since he had been the first to use the mark in the US; it didn't matter that he copied the mark from the Japanese, because trademarks, like copyrights, are not really international in nature, and use in Japan didn't count for anything in the US.
Third, your history is wrong. The Swift Company's divisions E.K. Pond and Derby started selling PETER PAN brand peanut butter in 1928, at a time when the plays were not only copyrighted, but Barrie himself was still alive and still held the rights. There's no indication that the name was used with permission, that there was a license agreement, or that there was any payment, or that Barrie even knew about it. Great Ormond Street Hospital, which wound up with the copyrights says that they never gave permission or got paid during the remainder of the time that the plays were still copyrighted in the US.
Had the character been called something that was simply descriptive of him, such as "Orphan Flying Boy"
You have a way too narrow view of what's descriptive in a trademark context. A mark is descriptive if it has an ordinary meaning and is used in conjunction with goods or services directly related to that meaning. Show anyone a picture of the Mickey Mouse character and ask them what that is, and they'll tell you that it's Mickey Mouse; the name of the character is a characteristic of it. If it weren't, you could say that no, the picture is of a character named Dan O'Neill, and people would believe you.
There is nothing magic or special about the US's copyright or trademark laws that would suggest that what would happen within the US should be even slightly different than what has already happened elsewhere when those jurisdictions have extremely similar laws.
No, there are two special things. First, many other countries regard international treaties such as the Berne Convention as being law which can be used by parties to a suit. The US does not; we regard most treaties to simply be obligations on our government to abide by them, but our actual law is what our own government enacts. Maybe they'll pass laws that comply with their treaty obligations, maybe they won't. We've been in violation of the TRIPS Agreement and the Berne Convention with regard to our homestyle exception (it concerns publicly playing the radio without owing royalties or needing permission) for well over a decade now and not only are we not about to fall in line, but we've deliberately gummed up the works so that we can keep on violating it and getting away with it. And it's totally legal in the US to rely on the homestyle exception. No one can sue you for using it in violation of treaty. This is because to Americans, copyright and trademark treaties are really only useful for 1) circumventing our own democratic institutions to impose greater protection for private interests, and 2) as a part of a trade policy meant to fuck over everyone else in the world, partly for those same private interests, and partly just because we can. There's no degree of amity or anything involved.
Second, we don't care about the rest of the world very much. We have our own statutes, and centuries of our own copyright and trademark caselaw. Only under extraordinary circumstances (or if we were interested in pre-American history) would we look at the law outside our borders for guidance on these subjects. Often our laws are somewhat different than those found elsewhere, such as how we're basically the only country in the world to have proper fair use. Whether or not Disney holds a MICKEY MOUSE trademark that effectively inhibits the free use of the public domain in Canada would not, for a single second, enter the mind of a judge who had to decide on how to treat the mark in the US when Steamboat Willie enters the public domain. Maybe this is a bad attitude for us to have, but it's par for the course around these parts.