--- Quote Start ---
Is it possible to write the 2 codes independently
--- Quote End ---
Yes, people can work separately but it's obvious that if they use NIOSII, they need also to be working on QuartusII, and NIOSII EDS.
You do not necessarily need to connect the NIOSII to the rest of the FPGA.
You can also use multi NIOSII on the same FPGA, every one for a different task. for Ethernet case with NIOSII, you need Nichestack wich is with License.
The link between HDL and C is the .sopcinfo file generated by SOPC or QSys. They give you the HDL file to add to your top level as well as the .sopcinfo that will generate a BSP file system to compile your code.