Yes you can use processing to do that.
You dont have to run processing to execute programs. You can export your sketch as standalone App : File - Export Application.

for your buttons, you can either use controlP5, great library for buttons, sliders, etc : http://www.sojamo.de/libraries/controlP5

or another library would be Gui4p . http://www.lagers.org.uk/g4p/index.html

You can also easily save what you have done as image using saveFrame, or as pdf http://processing.org/reference/libraries/pdf/index.html  but also export as 3d files if you want to http://processing.org/reference/libraries/dxf/index.html

If you want to save other stuff more related to your programm you can write your own file format, saving programm settings, coordinates etc. same with the loader. printing from withing processing was discussed here : http://processing.org/discourse/yabb2/num_1160674179_for_example_.html

I just checked out GEOGEBRA, doesnt look like an easy task lot of math... surface lib for example is a nice library that helps you to create 3d shapes http://code.google.com/p/surfacelib/
you can also add your own surfaces, maybe check out paul bourkes page http://local.wasp.uwa.edu.au/~pbourke/geometry/

good luck. and if you ask me, start thinking small and let your programm grow with your programming skills. maybe changing some simple graphs in 2d with sliders and buttons would be a good start