As far as I know (from listening in karate class) martial arts started in China and were brought to Japan along with calligraphy and art. However, the stereotypical, "karate" (there are many many different forms) was "invented" in Japan.
Shotokan karate was brought to Japan from Okinawa.
There are not many karate schools in China. The government there promotes Wushu. Karate is Japanese and the history between the two countries is not good.
China
Karate did not develop in China. It was developed in Okinawa based on a combination of wrestling and Chinese Kung Fu.
The karate kid
China and Japan are not continents but countries. They are both located in the continent of Asia. However, karate comes from Japan.
Karate translates as Empty Hand. Itcan also be translated as China Hand.
Karate is not Chinese. Karate comes from Okinawa, where it was created by combining Okinawan wrestling and Chinese Kung Fu. In China, Kung Fu is said to have originated in the Shoalin Temple.
No, Okinawa, Japan is the birth place of Karate. But somehow Karate is influenced by kung-fu ... so it depends
All over the place. A quick search on 'karate' and 'Minneapolis' brought up over a dozen schools teaching karate and tai kwon do.
The original Karate Kid took place in the United States. The new Karate Kid takes place in China.