Go and do the tutorials for your board. DE1-SoC has step by step examples on how to move data from processor and HPS.
Then go and read the Handbook of your chip to understand how the FPGA is seen by processor. Basically the FPGA is mapped in the address space of the HPS. So you have 2 options to move data to the FPGA. One is to use mmap (like in the tutorials for your board) and once you have a virtual address representing the FPGA you use that address and pointers to move data like any other peripheral. You can also use memcpy that is a standard function to move blocks of data and its faster that using pointers inside a for loop.
Other option is to use DMA in the HPS. I have an example here:
Aswering to your Questions:
-yes you have to use Qsys. In Qsys you add the HPS, otherwise it is inactive. And using Qsys you connect your custom logic to the processor. First you have to do a memory maped slave in VHDL and debug it through simulation. Then do a Qsys component from it to be able to connect it using Qsys. The connect your component and the HPS in Qsys. There are tutorials on how to do all that. Its impossible to explain here.
-yes you need the SoC EDS. It install the compilers for HPS. Using the SoC EDS you can compile the C/C++ for the processor. You can also use the DS-5 that is optionally installed after installing the SoC EDS.
-about the drivers I dont know. you have to connect it and see if thats possible.