Bows are shown strung at all times. Actually a bow would always be unstrung when not in use to prevent it from weakening and save wear on the string.
The movie is set in 1174 and has Richard I as King and away on crusade. Richard was not crowned King until 1189 and did not leave for the Crusades until 1190.
Contrary to the statement made by Robin's father, the office of Sheriff was not a hereditary position. It was an appointed office, usually assigned by the king.
The archers all carry what are clearly modern flat fiberglass bows, sometimes wrapped with cloth to try to disguise them.
According to his back story, Gareth cannot speak because his tongue has been cut out. Yet on at least one occasion you can see him lick his lips.
The forest setting changes often and suddenly. Many scenes are in a coniferous forest (not native to England but it is native to the Pacific Northwest) with obvious stands of pine. Sherwood forest is a deciduous forest.
As a member of the nobility, Marion would have had no expectation of marrying for love instead of political expediency.