Looking at things u seems to be new to GIS domain.I will answer to your questions one by one.
1)I suggest you go through Understanding_Map_Projections.pdf from ESRI ArcGIS s/w.(Google for it).It's 18MB ,so my internet speed would not be sufficient to send by myself.That should give you fairly good overview on projection stuff.
I suggest you use PROJ4 library(http://trac.osgeo.org/proj/).There is no need to reinvent the wheel,coz this is a stable,opensource,widely accepted library on projections.And I bet it's quite an effort to write things from scratch. This answers your Fifth&Sixth questions.
2)Your understanding(Third&Fourth points) on setting up glOrtho coods is absolutely correct.
3)As regards to your first&second queries,
a)Remember Lat/Longs spherical coods(Earth's shape) , are ANGULAR DISTANCES.But, we all are interested in LINEAR MEASURES.So it is the business of projection libraries(eg:-PROJ4,http://trac.osgeo.org/proj/) to do the conversion for us.
b)WGS84 ,alternatively called as WCS or Geodetic system is just going to provide us with Lat/Long.If u have no serious calculations involved and you merely require maps to be dislayed ,then there is no harm in using WGS84.
If one is interested in more scientific calculations,one would use suitable projection depending on the application area or Geographic region.For instance,US army uses MGRS system ,US Geological survey uses UTM/NAD and India uses Polyconic(Modified EverestProjection).
You may not know in which projection a shapefile is in,unless it is accompanied by .prj file with the same name,which holds projection information.You may use Shapelib(http://shapelib.maptools.org/) to parse shapefile.The following section gives us the shapefile boundaries.If the boundaries are between -180 to 180 and -90 to 90 ,then we may assume it to be UnProjected(WGS84).If you get some other values,u must read the associated .prj file to ascertain the Projection.
For eg:- SHPHandle hSHP=SHPOpen(fileName, "rb" );