homework and exercises - Can someone explain this solution for the motion of a falling chain?

In an example of Marion's classical dynamics 5th edition, I found example 9.2 not making sense, which states:

My questions are:

1. 
   

   The horizontal motion cannot be ignored even in the idealized case. As the velocity $\dot{x}$ tends to infinity, the horizontal momentum (in the direction of $y$) must grow to a certain value as $x \rightarrow b$, which implies $\dot{x} \rightarrow \infty$ (and in reality, $x$ and $y$ direction motion are kind of dependent on each other). In other words, if the horizontal motion will always be observed, then how can he assume no horizontal motion? Also since $\dot{x} \rightarrow \infty$, how can he not use theory of relativity?

   
   


2. 
   

   Energy is not conserved, since there is an additional tension force on the right side and doing work, compared to free fall. (In other words, the tension is contributing a certain amount of energy to $K$)

   
   



3. 
   

   After it comes to rest, where the heck does the energy go?

   
   


4. 
   

   An experiment done indicates that the tension force at A is 25 times that of the weight. Therefore I can't help wondering how much difference of order of magnitude can be treated as "infinity"? (order of one certainly not...)