Is the gairaigo メリー mainly used in the set phrase メリークリスマス (Merry Christmas!) or can it be used without the クリスマス to mean "merry", such as 私はメリーです to say "I am feeling merry"?
There's also メリーゴーランド (merry-go-round), but that's more like a transliteration of a single noun.
If it's mainly used in the set phrase, is there a term for loanwords used only in set phrases?
Note: The answer notes that メリークリスマス is not a set phrase after all.
In ordinary daily-life Japanese at the moment, メリー only means a person's name 'Mary' or 'Merry'. And even for 'Mary', it is usually written as メアリー, and メリー is seen only in a few established expressions like メリーさんの羊 'Mary's little lamb', メリーポピンズ 'Mary Poppins', or クイーンメリー 2 'Queen Mary 2'. Also 'Merry' is not much popular name, and is limited as in メリー喜多川 'Merry Kitagawa'. Other than that, メリー cannot be seen as a word in ordinary daily-life Japanese.
メリークリスマス is a gairaigo (crucially, not wasei-eigo) that originated from Merry Christmas, which means that you cannot divide メリークリスマス into its parts and make sense. The original English expression Merry Christmas can be divided into merry and Christmas, but dividing the Japanese word メリークリスマス into メリー and クリスマス is no different from dividing it into メ and リークリスマス; it does not make sense, and these parts do not have any terms by which they are referred to. クリスマス is another gairaigo that originated from Christmas.
メリーin the context of私はメリーです, it might be somewhat confusing to the average person. They might think you are saying, "I am Mary/Merry", as ifメリーwas your name. Perhaps something along the lines of私はメリーな気持ちを感じられますwould be more appropriate?