When I download and install the free Kazuraki font I do not get all the cool scripts just this:
or any of the other ones...
All of the cool script-like glyphs in Kazuraki are encoded in their appropriate code points according to Unicode, and you must thus use a Japanese input method to enter the characters, which is usually done via pronunciation. If that is not the issue, and if you copy the Japanese text above (書風をベースに、アドビのタイプデザイナー西塚涼子がデザインした書体です。西塚涼子は、藤原定家), paste it into an application, select the text, then apply Kazuraki as the font, and the correct glyphs still do not appear, it may be an application issue. It would then be helpful to know which OS you're using (and which specific version), whether you have Japanese enabled, and which applications you have tried.
Also, was the font free because it was a registration benefit for purchasing an Adobe application or application suite, or did you download it from some other source. If it is the latter, it may be a bootlegged version that includes only the Latin glyphs (which, by the way, are based on Brioso Pro Semibold Display, but scaled and shifted to better match the glyphs for kana and kanji), and it would be helpful if you were to send to me (email@example.com) the font file so that I can confirm this.
BTW, I was the one who did the production of this font, so I know all of of its particulars intimately well.