Designing 3D Furniture with Unity 3D? Is it possible?

It depends on what you're after, but everything you're looking for is theoretically possible yes.

If you mean creating furniture from scratch, (that is, defining and shaping vertices), you're better off using a 3D modelling software (Maya, 3DS Max, Blender) and then importing to Unity. If you want to though, you can also generate/edit a mesh by using Unity's Mesh class https://docs.unity3d.com/ScriptReference/Mesh.html . It's possible to replicate 3D modelling tools in Unity, but you have to code them yourself (or find an Asset Store package).

If you mean swapping parts of a model in real time (like character customization) yes, that's also possible in Unity. You can create instances of a furniture parts using Instantiate() and delete them using Destroy() https://docs.unity3d.com/Manual/CreateDestroyObjects.html There are overloads of these two methods that take world position and rotation. If you like, you could then get fancy and combine the parts into a single mesh also by using the Mesh Class.

For implementation, you could create a Monobehaviour with several lists of furniture parts, and then write methods to create/delete those parts from the world depending on what you want to display.

What you're asking uses some knowledge about how Meshes and Rendering works in Unity, so I would recommend doing some of the initial tutorials at https://unity3d.com/learn

You'll likely be using these classes, so looking into them might be helpful too, - Transform - Mesh - Mesh Filter - Mesh Renderer - Material

Here's a slightly dated tutorial on mesh generation, but it's still good: http://kobolds-keep.net/?p=33

Hopefully this helps!