I actually surf trough their webpages but seriously, I've never seen such a messy website and I didn't get how to apply for their scholarships.
Sorry but it was very hard to understand that you've already read the pdf files on that web pages and now looking for personal experience/suggestions.
Maybe you can ask one of the moderators to change the thread title to "Questions about JASSO scholarships" so that JREF members can get the right message and give you more appropriate answers.
As you said you've already read the pdf files, I'm very much sure you've already read this one which shows qualifications and how to apply very clearly but I'm quoting just in case you've missed it.
独立行政法人日本学生支援機構
Everything is explained neatly just in one page.
II. Scholarships by JASSO
1 Honors Scholarship for Privately Financed International Students
(1) Qualifications and Description
The qualifications for applicants and financial aid are shown as follows.
The qualifications do not include the age limit, designated Japanese schools to attend nor designated courses to major in Japan.
(2) How to apply
Applications for the Honors Scholarship for Privately Financed International Students will be accepted through universities, junior colleges, professional training colleges, colleges of technology or university preparatory course which applicants attend in Japan (hereinafter referred to as "schools"), submit it to their schools to have them recommend to JASSO that they are qualified to receive an Honors Scholarship. The applicants must contact the schools to confirm the application procedure.
The above answers your two main questions.
if scholarships are available for anyone or just for certain schools
(1) clearly sates that there's no limitation of the schools you attend - so any university you go, you are qualified to apply as far as your monthly allowance from family is under JPY90,000 and annual income of your finance supporter is under JPY5,000,000, etc. (See the details shown under "Others" of qualification.)
where should I go, what should I do,
(2) explains that application must be submitted through the school you're attending, which means you need to be a student attending a school in Japan.
If you intend to go to Kyoto Gaikokugo Daigaku and pay the fee by yourself, you'll be a privately financed international student.
So the answer is become a student of Kyoto Gaikokugo Daigaku first, then apply for the scholarship through the university.