For any advantageous trait that an individual member of a species possesses (for example, the very first "prototype" of the eye), it needs to have accumulated at least dozens of mutations that work together to produce that very specific affect.

Well the "very first 'prototype' of the eye" would be light sensitive cells. These cells were likely derived from the very first form of life: the single-celled organisms that metabolized light from the Sun. If you're asking us to explain how that evolved, you're asking us to demonstrate abiogenesis. We can't do that.

So as far as I can tell your argument is that since we haven't found a naturalistic explanation for abiogensis, God is necessary to explain it. This is unbelievably arrogant. Just because we've yet to find such an explanation, does not mean that one does not exist.

And on the topic of your apparent arrogance:

a robot can be trained to respond to stimuli, to move away from danger, but it cannot feel fear

How do you know this? Just because we've yet to do it, does not mean that it's not possible.

Also, can you guys concisely state the most convincing evidence you have for speciation as well as for the big bang?

There is no single piece of evidence in favor of speciation that is the strongest. What makes the evidence so overwhelming is the sheer amount of evidence from so many independent sources that all point, unmistakably, in the same direction. The most striking piece of evidence for me, however, comes from genetics. Using comparative sequence alignment, you can identify similarities in DNA sequences between different species. These similarities suggest a hierarchy: a family tree. My absolute favorite piece of evidence is the fusion of Chromosome 2.

The strongest, single piece of evidence in favor of the Big Bang is the discovery of the Cosmic Background Radiation. The best explanation we have for its existence is the Big Bang model.