I'm not sure about the first question, but one way I've found, though there's probably an easier way to center the unit, I just hit 'n' and then 'p', which in turn takes you to the next unit, followed by taking you right back to the unit you wanted centered, which centers it when selected anew like that
There is no need to deselect a unit.
If you need to check the terrain or an enemy unit on another hex just right click on that hex.
As Charles22 mentioned in his post in the general battle screen the use of the NEXT - PREVIOUS buttons is a good way to center the map on the currently selected unit.
If you are inside the artillery menu you may use the "F" (="Go to the current/spotter Unit").
You can also press "T". If there is no targets for the unit, it will just center. If there is a target, it will center on the target, but pressing "T" again will return to the selected unit and put that in the center of the screen.