This is just what I know (or knew) from my past experience. There may be others who have more up-to-date info, but here goes:
My questions are:
1/Is Cambridge A'Level certificate recognized in Japan university?
It may be used as proof of your English ability but it doesn't contribute to admission into a Japanese university (i.e., admission into university courses taught in Japanese).
2/Am I considered a Japanese or International Student?
You are Japanese but don't qualify for admission to most Japanese universities because you don't have education in Japan in Japanese schools. You belong to the category of Japanese who have had non-Japanese international education (such as at international schools in Japan). In this respect, you are an international student.
3/Is there any Identification Card in Japan that I need to apply?(I only have a passport and birth certificate)--->i found this question answered in wikipedia
If you have a Japanese passport, you are Japanese. No need for ID cards. The Japanese normally use their drivers' licenses as ID.
However, you may have to look into whether you are registered in your late father's koseki (family registry) in Japan to confirm this. The koseki is the definitive proof of your Japanese citizenship.
4/My birthday is on November and I will be 20 on that day. Since I'm not yet considered an adult, do I need any guardian to sign for me?
I don't know what you want to "sign," but you are legally adult on your 20th birthday.
5/I am not from a rich family and I need a financial assistance or bonds that could reduce the burden of education fee. Anyone that has some information on that?
There are financial assistance programs for Japanese citizens, but I don't know whether they apply to you since you're not eligible for "regular" Japanese university admission. You may have to check into such matters through your father's municipality in Japan, schools you hope to enter in Japan, etc.
6/I am thinking of teaching English and Chinese(private tuition) in Japan. Do I need to apply for anything?
Teaching English/Chinese is easy, so long as you can find students who are willing to pay, because you're free of visa problems.