Definitely no comma if you use 'ancient classical,' since the underlying term would be 'classical Arabic' (namely, the literary and liturgical language), and 'ancient' would be a subset of that.
However, the Wiki article doesn't list any category that exactly corresponds to that name, so you may need to do a little more research and be sure you're using a name that's well supported in English-language Arabic studies.
My own preference would be for lowercase adjectives, unless you are very sure that the complete collocation is a standard, widely accepted name for the language of a particular well-defined period (like Old High German or Middle English).
Wikipedia uses more capital letters, but it's unclear whether that's because the terms are accepted as standard, or because the first word is capitalized each time as the start of a subhead in an outline, or because Wiki often follows British style, where capitalization is used much more, shall we say, generously.
>>Hiǧāzī ist sowieso ein Eigenname, also auf jeden Fall groß.
Yes; and depending on your audience, you may want to consider the traditional spelling Hejazi, used e.g. in Wikipedia.