唐絵目利き kara-e mekiki ‘Chinese art inspectors’ - When Japan was keeping the outside world at arm's length during the Tokugawa era, the Shogun employed inspectors to appraise, catalog, and often copy samples of all goods coming from China and the West, perhaps as much to make sure the Shogun got the best goods as to keep harmful influences out. The characters that make up mekiki are 目 me ‘eye’ and 利 ki(ki) ‘efficacy, expertise’. But the latter also occurs in other contexts: ri ‘advantage, profit’; ki(ku) ‘to take effect, operate’; ki(kasu) ‘to use (one's head), exert (influence)’; ki(keru) ‘be influential’; and ki(kaseru) ‘to season’.
唐絵 kara-e ‘Chinese painting’ - Kara is written with the character for the Tang dynasty, otherwise read tō (< Tang), as in 唐画 tōga ‘Chinese painting’, a synonym of kara-e. However, 唐 means not just ‘Tang’ or even ‘Chinese’, but ‘foreign’, especially when pronounced kara- in native Japanese compounds, as in 唐行き karayuki ‘going abroad’ (lit. ‘Tang-going’), 唐草 karakusa ‘arabesque’ (lit. ‘Tang grass=flowing style’), and 唐黍 karakibi/tōmorokoshi ‘maize, Indian corn’ (lit. ‘Tang millet/sorghum’).
Compare the wal- (cognate with Welsh) on English walnut (once ‘foreign nut’); or the 胡 hu (once ‘barbarian’) on Chinese 胡桃 hutao ‘walnut’ (‘foreign peach’) or 胡椒 hujiao ‘black pepper’ (‘foreign pepper’ vs. 辣椒 lajiao ‘hot pepper’), or 胡麻 huma ‘sesame’ (‘foreign hemp’).
紅毛絵 kōmō-e ‘Dutch painting’ - By Tokugawa times, the Japanese had to deal with a new kind of foreigner very different from the Asians lumped together as kara. The character abbreviation for the Dutch is 蘭 ran (lit. ‘orchid’), short for Oranda ‘Holland’, as in 蘭学 Rangaku, ‘Dutch learning’, but by extension ‘Western learning’ more generally. So Western-style paintings can be called 蘭画 ranga, just as Chinese-style paintings can be called 唐画 tōga. But this book refers to the more specifically Dutch-style paintings from Nagasaki as 紅毛絵 kōmō-e ‘Red Hair painting’, a term I found especially engaging—as a former redhead myself (now mostly white), married to another former redhead (now more brunette with strands of gray), and the parent of a red-haired daughter.
By the way, Katsumori Noriko, whose chapter on "The Influence of Ransho [‘Western books’] on Western-style Painting" compares Japanese paintings copied from originals in European books imported through Nagasaki, starts by correcting the conventional history that Dutch-language books were banned between 1630 (the beginning of sakoku) and 1720 (during the reign of Yoshimune). She says (p. 99):
In fact, these policies applied only to Chinese translations of Western books. Books in Dutch, presented as gifts from foreign visitors, had been preserved over the decades in the shogunal library but were largely disregarded. When the bibliophile shogun Yoshimune opened his library in 1720, Japanese scholars had the opportunity to reencounter and study ransho firsthand.