learning-threejs | Code repository for the examples from the Packt book | Learning library

Three provides a huge range of options. These are just suggestions, your best bet is to read the Three documentation start point and find what suits you.

A mesh is just a set of 3D points and an array of indexes describing each triangle. Once you have built the mesh you only need to update the verts and let Three update the shader attributes, and the mesh normals

Q1. Use Three.Geometry for the mesh.

Q2. As you are building the mesh you can use the curve helpers eg Three.CubicBezierCurve3 or Three.QuadraticBezierCurve3 or maybe your best option Three.SplineCurve

Another option is to use a modifier and create the simple mesh and then let Three subdivide the mesh for you. eg three example webgl modifier subdivision

Though not the fastest solution, if the vert count is low it will do this each frame without any loss of frame rate.

Q3. Using Three.Geometry you can can set the mesh morph targets, an array of vertices.

Another option is to use a modifier, eg three example webgl modifier subdivision

Or you can modify the vertices directly each frame.

You can redraw the canvas by removing current scene scene.remove(mesh); and add create new mesh add into scene

Demo http://jsfiddle.net/sumitridhal/x8t801f5/4/

You can add custom controls using dat.GUI library.

If the Object can be written to localstorage it can just as well be exported as a file. You can send them to a server and store them there (maybe something like firebase would be useful here), or you can intiate a "download" directly from the browser. This is explained in Create a file in memory for user to download, not through server.

For loading a file, you can use the file-api, which is shown here: How to open a local disk file with Javascript?.

You just need to replace the localstorage-parts in your example accordingly.