There are actually two puns in this dialogue. First pun: Chat Noir: I would make a meat pun, but I'd probably butcher it. The pun is based on the fact that "to butcher" (verb) has more than one meaning. Literally, it means "to cut up an animal to make meat". "It also has the figurative meaning of "to ruin; губить": She really butchers the music. Она просто губит эту музыку. The humor lies in the fact that in the context of meat, both possibilities come to mind when you read or hear the pun. Second pun: Ladybug: Well, you know what they say about meat puns. Chat noir: Actually I don't. Ladybug: It's a medium rarely well done. "Medium" is often an adjective meaning средний. However, it is also a noun which has various meanings. One of those meanings is "a way of communication." (You have probably heard of the "mass media". "Media" is a plural form of "medium". The plural of "medium" is other "media" or "mediums."] In this pun, a "pun" itself is thought to be a way of communicating. It is a "medium" of communication. However, puns are "rarely (редко) done (рассказывать) well (хорошо)". The humor of the pun lies in the fact that we're talking in the context of "meat," and as you said, "well-done" also refers to how thoroughly meat has been cooked.

I don't read, or speak, Russian, so I'm not sure I translated your question correctly. However, in this narrative that you shared, Chat Noir made a pun (which is a type of joke). In response, Lady Bug answered with a clever pun of their own. This caused Chat Noir to jokingly ask Lady Bug to marry him/her, which was basically a humorous way of complimenting Lady Bug on their pun. So, basically, this narrative involved one pun on top of another. In case you're not clear on what a pun is, it is a joke which relies on the different possible meanings of a word, or a pun may use words that sound alike, but have different meanings. An English idiom that we use, which is a synonym for pun, is "a play on words".

глагол butcher имеет переносное значение. Посмотрите в словаре. Бифштекс "с кровью" будет по-моему rare. Да, Божья Коровка использовала все три термина. При этом It is rarely well done буквально занчит то, что сказала Лидия.

возможна следующая игра слов : "его редко хорошо делают" , т.к rare также имеет значение "редкий" , "well done " - хорошо сделанный ...