I have two questions:
1- Why is it important to have an different current path when operating a switch in series with an inductor? What happens if you don't?
2-The current in the inductor of the buck converter is not DC (not constant) but average output current is constant. What is the reason?

 

1 - If the feedback tracks also carry the switch current, you run a risk of instability, just like placing a microphone close to a speaker .
2 - The current in the inductor is switched on and off. That is how a SWITCH MODE converter works. The ratio of on to off varies the output.

Have you read up on how these power supplies actually operate?

http://www.learnabout-electronics.org/ is a good site for electronics.
And..
http://www.learnabout-electronics.org/PSU/psu30.php is the section on power supplies.

 

1- Why is it important to have an different current path when operating a switch in series with an inductor? What happens if you don't?

When the switch opens current needs to continue to flow.

https://www.google.com/url?sa=t&rct....mvp/id/2031&usg=AOvVaw0IodIZcKWLbOImdtvKOc1a

https://www.google.com/url?sa=t&rct...introduction&usg=AOvVaw0HELJSrSpaaJnMF2TPpUWl

Regards, Dana.