In Darwinian evolutionary theory, does the environment has any role?
"According to Darwin, eventually, the fittest will survive"

Darwin never said this. Fitness and survival are two separate concepts. 

In a simplistic version, fitness is measured by reproductive output. Survival is, well, survival. You can have higher output than your brother, but have a lower chance survival. In fact, the very act of producing offspring can negatively affect your ability to survive. 

No, the environment is not an other organism, in any context. It is the place (stage, arena, etc) in which the measure of fitness (reproductive output) can be measured. The environment can directly influence and affect fitness, and other organisms (inter and intraspecific) ARE part of that environment.

A single isolated organism does not evolve, but can have offspring. However, you need a more than one organism for fitness to have any meaning in a Darwinian model.