Removing the world map is fine. Star Ocean 1 did that and Secret of Mana also did that. It's neither a particularly good nor a particularly bad idea, but it's possible to justify it. I think without world map, the game's world will seem a lot bigger and easier to get lost in. I DO like that. Biggest disadvantage is that traveling around can get tedious, especially if backtracking is involved. The backtracking needs to be planned out really well and is usually solved by something like the transport cannons in Secret of Mana. I can also be solved by a "broad" map design, i.e. once you know the path, it's pretty short. Or by adding shortcuts (removing a boulder that blocked the shortcut and stuff).

As for dungeon design, I personally don't really mind maze-like labyrinths with lots of monsters. I grew up with Phantasy Star II after all. And I lost the booklet with the maps on.
I do understand how many players have the need for puzzles and variety, though.
I think it's all about the balance between the two. Only battles might made the game monotoneous (unless the battles itself are interesting, don't forget that battles could be puzzles in themselves too if well designed). But too many puzzles will make it feel more like a puzzle game.
Besides for puzzles you still risk to cause a lack of variety by simply making boring puzzles. I'd prefer bigger dungeon-themed puzzles over "slide these boxes to clear the room" puzzles any time.

Regarding equipment - both! There shouldn't be the need of having too many different equipments and at the same time the equipments should be fairly unique. Thought if you really want to focus on unique equipment needed depending on situation you might want to give the player a chance to change them mid-battle. Because keeping on dieing because of the wrong equipment being equipped before the battle is kind of frustrating.

Might be interesting to implement a simple, subway-like world map, with stops instead of free movement... like how Chrono Trigger or Final Fantasy Mystic Quest used the world map: adding in a simple way to get a good look at the world from a distance, and move to the next area. Something to consider if you've already started the work and don't quite want to abandon it.

If I remember correctly, the original Crystalis world was pretty big. And regarding Secret of Mana: it DID have a world map... accessible via summoning Flammie to fly to new areas.

Although, I'll defer to your personal vision.

Definitely. Great idea.

How did it work in the original Crystalis? One of those seers (Kenzu et al.) gave the player a Teleport spell?