This is the Pica 200 GPU demo. This is the 3DS GPU at work
Lambert, phong, specular, reflective, normals, Alpha, Environment and Cube Mapping. Baked and Dynamic lights are what I see in the demo; of course the sample environment is tiny so you can display all the goodies in 1 shot. You will be scaling things for larger environments.
To answer your question Occlusion, Global Illum *real time I assume?* and Displacement...or do you mean parallax? I haven't seen Parallax in hardware though I think it would be way to expensive to process per frame. I am doing some tests to see if it is possible...
Displacement like an animated sinewave effect ( vertex animated ) is possible. As for a 255 greyscale map ( bump map ) distorting geo "Displacement" The 3ds does have a tessellation shader. If you use the Pica 200 shader language I know it can be accessed. No idea if it works with bump maps, never messed with it. I know it would be a bit limited but the tech is in there.
For Unity 3DS sadly no to all..except maybe the parallax shader... I'll update this when I know more...
***update**
Yeah no Texgen so for Unity def no
Remember though that the 3DS renders the top screen twice at an offset to generate that 3d effect. That would means twice the Occ, Illum and Disp calculations per frame.
Some suggestions:
To get the effects you describe you can Bake occlusion into textures, Bake Global Illum - it will be static

. I'm old so I remember a world before global Illum existed which is why I say it can be setup and baked. Also remember that you have 3 active MB of Vram for the top and another 3 for the bottom screen. you will have to squeeze those textures alot. Most of the wishlist requires extra textures per shader + Mips. The rule with the 3DS is bake everything down and try to avoid per frame visual effects. I equate the 3ds to a PS2.5, it can do some limited console era equivalent effects but you going to optimize! Its going to make you boys strong!
Custom Engine and Normal Maps:
As for normals if you are using a proprietary engine, Normal maps are possible using The Pica shader language to make the effect. Mario Kart on the 3DS uses normals. Look at the games Banner "main menu 3d icon" and pay attention to the tires. Those are normals on the tires.
Unity and Normal Maps:
If you are using Unity the answer is ...sorta...You will have normal maps but they wont process in tangent space but rather in object space.
Voxel is right, dot3 bump mapping can be done via the texture combiner. but its not normal mapping the way the youth know it

. dot 3 does not interact with dynamic lighting since that is a per pixel calculation; not possible thanks to unity removing support for some fixed function shader functionality. Ie Tex Gen. Which from what I understand could be used to make a new matrix in the combiner shader.
https://docs.unity3d.com/560/Documentation/Manual/UpgradeGuide5-Shaders.html#:~:text=Unity 5.0 removed support for this fixed function shader functionality: UV&text=(dot3, dot3rgba). Any of the above will do nothing now, and shader
This ironically hamstrung Unity when it came to 3DS shader conversion and tangent space normal maps.

.
This ancient text talks about what Voxel mentioned using the Dot 3 Bump - if you can translate it to ShaderLab it works
https://www.oreilly.com/library/view/iphone-3d-programming/9781449388133/ch08.html
Lastly know I'm no expert. All this is my experience. I'm sure I have information gaps. I'm still learning but I'm happy to share some of these rabbit holes I've looked down