My Answers:
1. It depeneds. If it behaves like something else, and will run the same programs as the other thing in the same way, it is an emulator. If you created a system that runs code, but not in the same way, it is a simulator. If it is a brand new design, for example, an implementation of an FFT, it is neither an emulator or simulator.
2. I dont understand the question.
3. See 1.
Simulation usually means you have created something that runs like something else, but not in the same way (ie. input and output match, but functionality doesnt).
Emulation means everything matches.
Often, FPGA designs are not trying to copy anything.