You can mark your most used actions under Progress and Quality as favorites for quick reference in the Favorites window instead of having switch between the Progress and Quality windows.

Mine will automatically re-open for each new craft unless I closed it on the last craft. If you're asking for it to stay open while you're on the crafting log between crafts, probably not. It wouldn't know what numbers to display since those change based on recipe level/difficulty.

That goes back to what was mentioned with Favorities, With keyboard/mouse, it's a simple click with the mouse to switch between the windows. I'm not certain how a controller user would do it.