How to make Finder remember its window size and column sizes?
The simplest answer to 'how' rather than 'why' is:

Opt[alt] drag a column sizer. That size becomes the new default, after you relaunch Finder. Only works for column view, not coverflow.

Other options, though each is temporary   


Double click a column sizer to set that column to show full content width.
Right-click a column sizer for more options.


To relaunch Finder, hold Opt and right-click Finder in the dock. Relaunch will be in the context menu.