Should "japanese" be capitalised when used as an adjective

In English country names, and their associated adjectives and languages are always capitalised.

In some languages, such as French, only the country name is capitalised - the language and people are not.

Generally speaking, nations and nationalities are capitalized. This is always the case with things, like cuisine or history, that are closely associated with the the country. Thus Japanese cuisine (not japanese cuisine) and Chinese dynastic history (not chinese dynastic history).

There are a small number of exceptions, when the item described has a more remote connection. Thus

We'll use the good china

(not "the good China") to describe porcelain or vitreous dinnerware. And

japanned furniture

(not Japanned furniture) to describe black lacquered furniture. Opinion is divided on whether to capitalize english to describe the spin of a ball.

As per the capitalization rules set out in GrammarBook.com

Rule 2. Capitalize proper nouns—and adjectives derived from proper nouns.

Examples:
the Golden Gate Bridge
the Grand Canyon
a Russian song
a Shakespearean sonnet
a Freudian slip

In any Grammar book it shall be clear that sentence #1 – "Your favourite Japanese restaurant" – is the correct usage.

According to guidelines in the Chicago Manual of Style, the point of capitalising an adjective is to convey the literal relationship to the proper root noun. So, if you're trying to say that the food comes from there, or the restaurant is in Japan, or is staffed by Japanese persons, or is owned by a Japanese owner, or anything that literally ties it to Japan in a material way, it's appropriate to capitalise the adjective. If you're simply describing the style of the cuisine, a lower-case adjective is acceptable.

Japan (the country) is capitalized. And the adjective form Japanese.
Japanese (the language) is capitalized.
But japan (the hard black varnish) is not. Which means it is legal to use in Scrabble!