A few of my favorites.
Lunar 2, Eternal Blue. (Mega cd, Saturn, PS1)
Decidedly old school, but in the best way, solid battle system and a great sense of adventure and genuinely good character development. The translation for the PS1 version was top notch and often amusing. By the end it'll have given you genuine feels. One of my fondest ever gaming memories.
Chrono Trigger (SNES, PS1, PSN, IOS, DS, Virtual console)
What can I say?, it's fucking Chrono Trigger! imo, the pinnacle of the jrpg genre and others will describe it better. The PS1 and PSN versions suffer from loading times and are considered the worst bet, but did have added animated cutscenes. Personally I didn't think much of them, but others seem to like them. The ds version is considered by many to be the best, although the added content was completely throwaway, they also retranslated it, which is a deal breaker for me as I played it first on the SNES, but if you haven't played it before, it shouldn't be an issue. The virtual console version is my current favorite way to play, SNES faithful emulation in 480p on my 50 inch plasma!
Grandia (Saturn, PS1, PSN)
This game was very much in Final Fantasy VII's shadow when it was released, but it's easily one of the best in the genre. You want an adventurous feel? This game has that in spades. I also think the Grandia battle system is still to be bettered by any other jrpg and had some great subtleties and nuances. The english translation wasn't the best (but not awful) and the english voice cast isn't great(again, not awful), but don't let that dissuade you. I also recommend the excellent sequel that was released on the Dreamcast and PS2 (no story relations)
Secret of Mana (SNES, Virtual console)
SNES classic with a real time battle system and a good sense of advancement as the characters level up. Gorgeous colorful graphics and one of the best ever 16-bit soundtracks, it's a game I replay regularly. The dialogue's a bit ropey in places, but it doesn't effect the enjoyment in the slightest. It also has multiplayer where if another person grabs a pad, they take control of one of the cpu controlled allies, and the virtual console version even allows you to play three player games that would have required a SNES multi tap back in the day.
Shining Force 2 (Mega Drive/Genesis, Steam, countless Sega game compilations on various systems )
Bigger and better than it's fantastic prequel, this strategy jrpg was a huge feather in Sega's cap back in the early 90's. It plays along the same lines as Fire Emblem, Advance Wars, FF Tactics, but with it's own feel. Tons of characters to recruit to your army including fighters, mages, archers, centaurs, bird people, even a robot, with many of them hidden. Solid story and characters and sublime gameplay, it's battle system is where it excels. A game that's remembered fondly for good reason.
Dark Chronicle, aka, Dark Cloud 2 (PS2)
Pushing the boundaries of the term "jrpg" here, but i think it fits. Unfairly billed as Sony's Zelda because it's absolutely fuck all like Zelda.
I think this one ticks all four of your desired categories. Some may say it's a bit kiddish, but that's bullshit as it's just a game made before the current trend of making everything dark and moody, and the story definitely has emotional highs and lows. It's essentially a dungeon crawler at heart, but there's just so much other stuff to do as well, and the best thing is, none of it's filler. With inventing things, Weapon creation and upgrading, town building, fishing, golf, and every dungeon level having some sort of optional completion challenge to strive for if you want to go back, the amount of stuff in the game is crazy.
The game has quite a sedate pace, and it can be grindy to some, but I personally like that about it, I like to return to this game when I'm feeling burned out by more modern offerings, and I actually find the gameplay flow to be genuinely therapeutic.
Well voiced characters, who all have believable motives (no amnesia!) and a pleasant soundtrack. The cell shaded style of the graphics also mean that the games visuals have aged really well and it's pleasing on the eyes too.
Some of these games may not be the easiest to get hold of and play, but I reckon they're worth the effort and there's always emulation if you have that option.