Plants don't necessarily need extra oxygen because they can do photosynthesis. In photosynthesis, water and carbon dioxide (CO₂) are combined to make sugar, with oxygen as a waste product. This process requires energy from light. When plants don't have light, they do the same thing we do. They break down the sugar to release energy. This requires oxygen. Water and CO₂ are the waste products. As you can see, these processes are opposites. This means that plants can survive in a closed system as long as enough light is provided.

You can try this on your own by making a terrarium. You can use a big glass jar, a 2-liter soda bottle, or an aquarium. Someone may have a leaky one that you can use. Put in some plants with soil and some water. Cover it with plastic wrap or a clear plastic cover. There are a lot of directions available online for building different types.

Theoretically, possibly, but plants can and do need to use oxygen for respiration when they are not photosynthesizing (which they aren't at night). Most plants do this gas exchange with the air, just as animals do, so putting a plant in an airtight space will cause it to asphyxiate.