What is the law if a person takes them randomly, one piece at a time, without any order, and puts them back in placeis that permitted?

Some later authorities wrote regarding silverware that if one takes them one by one, it is permitted to place each one in its compartment. I am a bit doubtful about this leniency, but there is authority to rely on (since in any case this is only rabbinic-level selecting/sorting. Especially since these are large, easily distinguishable items).

As I recall, this leniency is applied with two people. Then it makes sense: one takes them out randomly and the other places them down. From the standpoint of each individual person, it is not an act of selecting/sorting, although perhaps the situation has to be viewed as a whole?
For one person, there is really a change in entropia.

Likewise, it would also be possible to spread them out far from one another until there is no mixture there in the spread, and then the second person arranges them.
But as above.

That is a case of two people doing it together, like one uproots and the other places.

Why is this selecting/sorting only rabbinic and not Torah-level? (Because it is a labor not needed for its own sake?)

Because it is not done with a utensil, and here one is sorting food from food. And there are opinions that selecting/sorting applies only to things that grow from the ground. Beyond that, we are dealing with large, easily distinguishable pieces. This whole issue is rather doubtful, but the accepted practice is to be stringent about it.

It is worth noting that it seems the Rabbi later retracted this responsum. See this link:
https://mikyab.net/%D7%A9%D7%95%D7%AA/%D7%A9%D7%99%D7%9C%D7%95%D7%91-%D7%9E%D7%A9%D7%97%D7%A7-%D7%A0%D7%95%D7%A1%D7%A3-%D7%9C%D7%9E%D7%A0%D7%99%D7%A2%D7%AA-%D7%91%D7%A2%D7%99%D7%99%D7%AA-%D7%91%D7%95%D7%A8%D7%A8-%D7%91%D7%9E%D7%95%D7%A0/

More power to you.