When Verena uses Close Quarters and the figure whose dice pool she is using is Focused, does she get the green die in addition to the figures normal dice?

Focused does not modify a figure's attack pool. Focused adds the green die when the figure having the condition declares an attack (or an attribute test).

So, no.

Besides, when the figure is defeated, it is removed from the map, drops all carried tokens and loses all conditions (and for the last figure in a group the non-unique deployment card returns to the imperial player's hand). So, the figure is no longer focused when Close Quarters is performed.

Thankyou