Well, in theory "Best Picture" is the best film of the year - a combination of every sub category, including director, editor, actors, sound, etc.
Which means that it's entirely possible the director (editor, actors, etc.) of the best picture of the year had a weaker entry than the best director (editor actors, etc.) nominees but as a full package, the combination of all those hundreds of people's in a films credit listings, resulted in a better movie overall.
Unless somehow the director has it in their contract, no, they don't have control over home video releases. Most don't even have full control over theatrical releases