Show entries in the Control Panel as a grid, to reorder position
In the Craft backend, I don't think this is possible without (creating) a plugin or module of sorts. Even then, I don't think the element index page is extendable enough to accomplish what you want to accomplish.

Even if you were able to get this done, one way or another, how would you take into account a responsive situation, where more or less than 5 image squares are shown for larger or smaller screens? Would the CP mimic this behaviour?

The only thing I can think of is some sort of table field (Super Table, maybe?), in which you have 5 columns and any number of rows, and every column contains one asset field.