The kinetic energy of the segments which come to rest is transferred to the moving part of the chain. This is the non-intuitive part of the experiment. Here is a video from Veritasium chanel, which explains it.
For real chain as soon as the last segment reaches the bottom the horizontal motion is no longer neglectable. The last segment of the chain starts rotating and there is no next segment to which it could transfer energy so it starts swinging. The swinging causes the whole chain to wobble. The enegry of this wobbling motion then dissipates. Please keep in mind that the example is only aproximation. The aproximation is good as long as there are many segments on the right side.
I took $b=10m$ and length of segment $\Delta x=5cm$. When the last segment reaches the bottom of the chain it has traveled distance
When I plug in equation 9.17 I get the speed of the last segment of the chain:
It's pretty high but certainly not relativistic.
The effect is important for function of whip. Many people think that the sound of whip is caused by the whip slaping itself. The reality is much more interesting. The end of the whip actually reaches supersonic speed!
Answers to your questions:
1) The horizontal motion can be ignored only until the last segment reaches the end. This is long enough for us to observe the character of the tension.
2) The tension force compensates for the decreasing mass of the moving part. The speed of the moving part increases compared to free fall but total energy is conserved.
3) The energy dissipates during the wobbling of the chain. At this time our equation no longer make sense.
4) The number is only there to compare with free fall. For free fall the tansion would be only one times the mass of the chain. If you take "more ideal" chain the number is going to be higher. You can improve your chain and still get higher and higher tension. There is no limit for ideal rigid chain and that is the sense of the infinity. Of course real chain brakes at some point.
This experiment is amazing and it blows my mind everytime I think about it. It actually took me 3 hours to write this answer.
Why doesn't he count for relativity? And why doesn't he count for the chain breaking, why doesn't he describe wobbling of the chain? Maybe because it's beyond the sense of the example