In short, yes.
Okinowan Martial Arts were eventually popularized by Gichin Funakoshi, who developed the Shotokan karate style. He took his style of martial arts on tour throughout the country of Japan, demonstrating them at colleges and other community hubs. Eventually many school adopted karate as a part of their physical fitness programs and many colleges have Karate clubs. It wasn't until after the second World War that karate spread out of Japan into western countries. By then it was "all the rage" throughout Japan.
Empty hand is Karate, a Japanese martial art.
Sport karate is geared towards competition rather than as a martial art. Budō is simply a Japanese word for martial arts, with gendai budō referring to styles that came about after the Meiji restoration - karate, aikido, judo, kendo etc. As such ALL karate is "budo karate".
Sumo wrestlings would be a popular Japanese sport though that popularity has been on a decline. Judo, Kendo and Karate may also count though they double as a martial arts.
No, karate is a unisex sport. It is made for both male and female.
karate is Japanese
No, Karate is not a cruel sport, no it teaches people how to defend themselves and have fun.
It is a martial art, but is somewhat different, but Judo and its root art of Jujutsu are Japanese martial arts.
Karate is a Japanese word.
Taekwondo is considered a form of karate and is the national sport of South Korea.
Karate is not an Olympic sport.
Karate is a Japanese world, so it would be said the same way. The Japanese word for child is kodomo.
karate!