It is more than that, it is essentially the entire clock containing the primary oscillator and dividers respectively counters. Watch closely, when several digits change at once, then it emits multiple pulses. Also note that the pulses are emitted from different places, there are essentially three counters, 0 to 12, 0 to 5, and 0 to 9. The three structures above the display are essentially just seven segment decoders, you can see the digit patterns encoded in them similar to mask ROM. There is one column for each digit and two rows for each segment, one to turn the segment on and one to turn it off. If you zoom in on the right of a decoder, the rows are actually labeled with [S]et and [R]eset and a letter A-G for the segment controlled by that row.
Looks like it, it sends out a "glider" once every second.
Why is this incredible? (Do you know "life in life"?) Is it the way the website is built?
That's nothing, have you seen CERN's LHC?
"Life in Life" is actually Conway's game of life... Directly comparable to this, arguably a lot cooler and not a new thing at all. "Incredible" should mean, at least to some extent, you didn't think the thing existed or was possible.
I'm not sure what you're trying to say. Is it "never ask anyone why they find something incredible"?
"Incredible" is commonly used synonymously with "amazing". Even though this isn't literally unbelievable (the primary meaning of "incredible"), it certainly is amazing.
It's also relevant to note that "amazing to the point of not being believable" is only the primary literal definition, while you would have a legitimately hard time finding even a single colloquial use of the word that includes the "not being believable" part. A similar case is "awful", which nobody uses to mean "full of awe".