In the video, at around 19:16, the UVs for the leather belt was amended by adding more seams to make the islands fill the whole grid in the image editor. However, when I tried doing that I was really unable to make the islands larger to fill the whole grid. Is it 100% necessary to make it fill the whole grid?
Also, when we mark additional seams, do we need to move the islands that were affected out of the grid before averaging the island scales and then packing them again?
You can always scale them to fit the zero to one space. I wouldn't call it necessary, but it is better if you used all the texel density available in your space. It is wasted if you leave your islands small and then your texture will be lower resolution that it should.
You don't need to move the islands, the packing algorithm will move them for you in the end, so I'd say it doesn't matter where you place them if you're going to pack them.
You don't have to make them fill the whole grid - it's just that the more space they cover the higher the resolution of the texture will be as you give the model more pixels to work with.
You can scale and rotate the UVs just like you can with geometry to fit the grid better and then PIN them so when you repack the other UV islands they won't be affected.
You don't have to move the islands before averaging and packing again. It just helps to see what you're working on and makes selection easier.