Icon order in toolbar flyout

Yes, you can.

My personal recommendation is to start by going to Options>Workspace and creating a new Workspace to experiment with and become familiar with workspace customization.

There are at least two ways to move Commands around:

• With Options>Workspace>Customization>Commands open, one can drag-and-drop to move Commands around within toolbars/menus, move them between different toolbars/menus, remove them from toolbars/menus, etc.
  
  Here's an example where I create a new workspace starting from the Default workspace, then move Contour to the top of the flyout and Blend to the bottom. Then, I switch back and forth between the Default workspace and the new one to see the changes in the flyout.
  
  VIDEO: moving Commands

• Without having Options>Workspace>Customization>Commands open, then one can still do these things - but it requires holding down Alt before doing the drag-and-drop.