Are names of Languages proper nouns? Should they capitalised?

Yes, in English we capitalise the names of countries and the adjectives formed from them. But we don't capitalise the word "language".

She speaks good German, but her French is better. Bob's knowledge of Asian languages is astonishing, and he speaks Thai really well.

They are proper nouns, so they have be capitalized.

yes, always.                            

Yes, the names of languages, in English, should be capitalised. This is so even if the word is used as an adjective. You may say, She speaks the Portuguese language.' In that sentence 'language' is a noun but begins with a lowercase letter. The word 'Portuguese' is an adjective but still begins with an capital letter.

Your examples are correct, and refer to all names of languages, whether used as nouns or as adjectives. 

 In most (all?) other European languages they do not start with capital letters.

Yes.

The names of languages, like the names of countries, are proper nouns and should be capitalized accordingly.

Please also note that several of those languages (like English and French in your list) are nationalities as well.  Which are also proper nouns.