Astronaut is the original English word. Cosmonaut is a transliteration from the Russian word for astronaut. So if we're speaking English, we should use astronaut regardless of the country they're from. Otherwise we're forced to invent silly new words that nobody understands like taikonaut and what will we call Indian astronauts or whoever's next?
"Astro- a combining form with the meaning “pertaining to stars or celestial bodies, or to activities, as spaceflight, taking place outside the earth's atmosphere,” used in the formation of compound words:"
It sounds like the Greek(?) meaning has been lost now in English, which is fine since it's a different language. Astronomers still sometimes look at other planets afterall.
Not obviously. If we're going to quibble over details, we might as well do it properly. The astro- root can also translate as a non-stellar celestial body, since at the time the word meaning was established, the planets were considered "wanderer" stars. Also, asteroids are not stars, but bear a related etymological root.
Cosmos, on the other hand, derives from the entire universe. Obviously, the universe surrounds us completely, but if you're willing to call someone in LEO a cosmonaut, I'm not certain you couldn't say the same about someone driving a car on the planet's surface to the grocery store to buy milk. Both are traveling through the cosmos. Where's the cutoff? Earth escape velocity? Solar escape velocity? Galactic escape velocity? People in LEO just aren't going fast enough.
I have no qualms about calling the flight personnel of the Apollo program "astronauts", as their goal was to reach another celestial body, even if it was not literally a star, and several of them actually made it there. "Selenauts" would also have been appropriate.
But since 1972, we have only been sending humans as far as LEO. Maybe "lacunauts" would be better for those in space, but not traveling to other celestial bodies?
Astronaut translated to Chinese is 太空人 which is spelled by the romanisation system in mainland China as tai-kong-ren or taikongren. Not the extra 'g'.
A quick search of matches in actual translated texts returns 'astronauts' as the term in the vast majority, and a few cases of 'spaceman' and 'man in space'. 人 'ren' in Chinese is genderless.
Taikonaut is however used by the Global Times and China Daily, tabloid and broadsheet papers published in the mainland. Am pretty sure the term wasn't termed in China, however. Otherwise the pinyin would be correct.
Cosmonauts = Russian Taikonauts = Chinese Astronauts = Western