Things that are glowing, like beams of energy, don't have shading. But, you are on the right track with using something like pillow shading for them. It's just that instead of going from midtone to shadow, you should be going from bright to less-bright. It might also feel more like a glow if the bright part dominates the beam, with only a few darker pixels. Also, where the beams meet, there should probably be more intensity, instead of the beams fading out towards "shadow" before they make contact.

Here's an edit with all of my suggestions:

Yes, that's what I meant! Sorry for not making it clearer.

You have shaded each segment in a way that shows its shape, but I am not sure that it's conveying the creature's forms overall. If the light is above, should the back calf not have more light on it? Shouldn't the textured area in the middle have more light? Since this isn't a real creature I don't know what it's meant to be like, so I'm just guessing at what might help make it look more 3D.

I don't think you're going to get that looking good at this scale without adding animations for running to the side, or even better, 8-directional animations (with mirroring, that's 5 different animations). Why should the character only have two running animations?

Many level editors support adding objects in this way (it's the same process as creating/placing NPCs, the only difference is they have no scripts or have different types of scripts associated with them). It is rare to find engines that support tile-based non-grid "sprites" out of the box these days; it always seems to require custom code, but it can be done.

The reason it isn't done more often is because it creates a performance hit compared to doing everything grid-based, since it requires more sorting (per FRAME as different characters weave between the static objects) and more pixel-position calculations as the camera moves. You'd think that with modern computers the performance on 2D games shouldn't be an issue, but it still can be, especially if you're making a browser-based or mobile game. Also, for a well-designed tileset and map, the quality difference is often not that noticeable.
And of course, grid-based levels reduce the complexity of the code and the work a level designer has to do.

World tiles can be animated, even dynamically (e.g. to react to wind caused by a player's attack), so for something like wind animation, there's no need to turn things into game objects unless there's another factor requiring it, e.g. if you really want to erase the grid beyond an artist's capability, or those objects are also something the player can walk around and that you can't handle with clever layering. Keeping static objects as part of the world layers, even if they're animated, helps with performance since it avoids unneeded depth-index calculations and reduces the number of draw calls.

Truly interactive objects, such as piles of rocks that can be shoved apart by a player attack or by walking into them, should of course be game objects. Whether they're grid-aligned or not depends on the look you want, but assuming you want a very naturalistic world, why not have them off-grid? That is already how it's done in most games I think, although many keep their interactive objects grid-aligned for aesthetic and gameplay reasons.