Like other language names in English, this one is uncountable and takes no the in ordinary sentences: speak Danish, learn Danish, teach Danish. When you want to say which language something is in, you use in Danish. The capital D stays, because the name of the language is built on the name of the country.

This is a label you can put on almost anything that comes from Denmark: a person, a town, a chair, a cheese, a company, a newspaper, a word. It always keeps its capital D, because it is built on the name of a country. It has no comparative, so you do not say more Danish or the most Danish in ordinary use. For a person from Denmark, the usual noun is a Dane, though a Danish person is also fine.

This one is food, not a nationality, and it is the only use of the word you can count: a Danish, two Danishes. The short form is informal and chiefly American, and both spellings turn up in print, with a capital D and with a small d. In a cafe the flavour usually comes first: a cheese Danish, an apple Danish.