Conducted on the same day of Graduate School Entrance Exam: Once for Medical Sciences, and Twice for Pharmaceutical Sciences as of FY 2015.
Current application status can be checked by visiting;
http://higoprogram.jp/enhowto_apply/

You can. If there still remains room after HIGO Selective Exam conducted in April, another Selective Exam would be announced.

12 selected students are eligible to receive HIGO scholarship. Eligibility can be extended to the following fiscal year if qualified by assessment. HIGO scholarship will not be offered to the students who are receiving other grant(s). Please make sure the amount of scholarship might change due to budget conditions. The nation supports the Program until FY 2018, which might not ensure HIGO Scholarship after April, FY 2019.

You need to decline the grant once you are qualified for HIGO Scholarship, and can apply for the grant or other kind of stipends when HIGO Scholarship ceases to be provided.

You can.
About admission fee exemption/moratorium, see;
http://www.kumamoto-u.ac.jp/daigakuseikatsu/nyugaku_zyugyou/nyugakuryo_menjo
About tuition fee exemption, see;
http://www.kumamoto-u.ac.jp/daigakuseikatsu/nyugaku_zyugyou/jugyoryo_menjo

KDS (Kumamoto University Doctoral Course Scholarship Program) and KWS (Kumamoto University Ph.D. Work-Scholarship) are not applicable for HIGO Scholarship recipients. If you are already supported by KDS or KWS, the service should be stopped once qualified for HIGO Scholarship.

All the HIGO Scholarship recipients are not allowed to earn money by doing any kind of job. If you are already working, you should retreat from the job by the month preceding Scholarship commencement.

Can. Although you have to give up HIGO Scholarship, your place at HIGO Program can be secured.