The next five games on my master list fail one or more of my criteria, so I'll cover them all here.
Soul Blader: Released in the US as Soul Blazer. I loved this game as a kid. It was a fun action RPG, and it was easy and short enough that I could rent it and beat it over the weekend (or at least play it enough to have fun). Releasing all the souls was always a fun experience as you see the world build up, and I remember it having good music as well.
It's the first game in what's often considered a loose trilogy; I love all three games -- Illusion of Gaia had a memorable story, and I think Terranigma is one of the best Super Nintendo games.
Zenic Reverie has already covered this game.
Dragon Slayer: Legend of Heroes: This is a port of a computer game; like many of Falcom's games it was ported to a bunch of different systems. I'm a huge fan of the Kiseki (Trails) series so I would like to play this at some point, but I'll probably play the PC Engine (Turbo Grafx) version. Zenic Reverie did this one as well.
Super Inindo Datou Nobunaga: Released in the US as Inindo: Way of the Ninja. This is also a PC port, and was released in English. A friend of mine had it and I remember him not enjoying it, but I don't remember that much about it.
Zenic Reverie is currently playing this.
Card Master: Seal of Rimsalia: Released in the US as Arcana. I remember playing this as a kid; it was just as a rental so I never finished it, but it's one of many first-person dungeon crawl games.
Zenic Reverie did this one as well.
Ultima VI: The Ultima games have a lot of bad console ports, and although I've never played this, I can't imagine it's any good. The PC game is great; I've always been a huge fan of the Ultima series and I liked VI's story even though there were a lot of flaws in the gameplay.
CRPGAddict did the computer version of this.