Has Anyone Created a 3D Website That Works on a 3D Monitor

3D object modeling that can change whenever the option is being clicked

To create the 3D models you can use several different programs. Autodesk could be one, blender... However if you have no idea and this is for academic purpose, you may find free 3d models.

Then to create the website you can use Unity as yous instructor say, however, not sure if it is the best approach. If you decide to continue with Unity, you can add a botton which instantiate a 3D model somewhere, for example in the center of the screen, when you click it.

I will leave you here several links you may read, and try things.

For 3D models:

https://www.assetstore.unity3d.com/en/
https://www.turbosquid.com/

To work with buttons (In general to work with the UI) in Unity

https://unity3d.com/ru/learn/tutorials/topics/user-interface-ui

To instantiate the 3D models, once you download them and you add them in your scene:

https://docs.unity3d.com/ScriptReference/Object.Instantiate.html

To publish your scene as WebGL:

https://docs.unity3d.com/Manual/webgl-building.html

Try this things, and come back with your progress

3D library recommendations for interactive spatial data visualisation?

I have used GLScene in my 3D geomapping software and although it's not used to an extent you're looking for I can vouch that it seems the most appropriate for what you're trying to do.

GLScene supports terrain rendering and adding customizable objects to the scene. Objects can be interacted with and you can create complex 3D models of objects using the various building blocks of GLScene.
Unfortunately I cannot state how it will work with millions of points, but I do know that it is quite optimized and performs great on minimal hardware - that being said - the target PC I found required a dedicated graphics card capable of using OpenGL 2.1 extensions or higher (I found small issues with integrated graphics cards).

The other library I looked at was DXscene - which appears quite similar to GLScene albeit using DirectX instead of OpenGL. From memory this was a commercial product where GLScene was licensed under GPL. (EDIT - the page seems to be down at the moment : http://www.ksdev.com/index.html)

GLScene is still in active development and provides a fairly comprehensive library of functions, base objects and texturing etc. Things like rotation, translation, pitch, roll, turn, ray casting - to name a few - are all provided for you. Visibility culling is provided for each base object as well as viewing cameras, lighting and meshes. Base objects include cubes, spheres, pipes, tetrahedrons, cones, terrain, grids, 3d text, arrows to name a few.
Objects can be picked with the mouse and moved along 1,2 or 3 axes. Helper functions are included to automatically calculate the top-most object the mouse is under. Complex 3D shapes can be built by attaching base objects to other base objects in a hierarchical manner. So, for example, a car could be built using a rectangle as the base object and attaching four cylinders to it for the wheels - then you can manipulate the 'car' as a whole - since the four cylinders are attached to the base rectangle.

The only downside I could bring to your attention is the sometimes limited help/support available to you. Yes, there is a reference manual and a number of demo applications to show you how to do things such as select objects and move them around, however the reference manual is not complete and there is potential to get 'stuck' on how to accomplish a certain task. Forum support is somewhat limited/sparse. If you have a sound knowledge of 3D basics and concepts I'm sure you could nut it out.

As for Firemonkey - I have had no experience with this so I can't comment. I believe this is more targeted at mobile applications with lower hardware requirements so you may have issues with larger data sets.

Here are some other links that you may consider - I have no experience with them:

http://www.truevision3d.com/
http://www.3impact.com/
Game Development in Delphi

The last one is targeted at game development - but may provide useful information.

3D models on HTML page

Chrome Experiments has released several 3D Javascript applications. Here's an example:
Colors Cube

How is the way to write 3D Web Application for Custom Tailoring?

The tailor application you gave a link you uses quite low-tech approach. It uses (pre?)generated images for each option (using transparency layer). Images are stacked one upon another to produce the final result.

Of course, the end-result is looking nice, which is due to careful web-design.

In short: you need a pile of carefully prepared images (it's a lot of routine work unless you have them readily 3d-modelled) for each detail and clever URL-mapping to choose the image. Everything else is just programming choices in HTML/javascript and loading correct images into corresponding layers.

Use browser's development tools (like Firebug for Firefox) to discover how the application works.

XNA Game Studio 3D model editor suggestions

Take a look at trueSpace also, its just become free - but as Evil Activity stated; Blender is also a good sugestion i just never really got use to its interface and how to do stuff. trueSpace is a little more easy in that way, but i think that blender is more powerful.

I guess you know http://creators.xna.com/, there is a 3D tutorial you can look at here:
http://creators.xna.com/en-US/education/gettingstarted

Going from Rhino3D to the browser

Here are some thoughts from a classmate of mine:
So three.js is where you want to start: http://threejs.org/
Download the library and check out all of the 3D model examples. Once you get the hang of that you can export from rhino and get it imported in webGL. The simpliest model format to work with is Collada. Rhino doesn't export this, but Maya does. I'd play around with the best workflow to get from rhino to webgl.