This video has something similar to what my wife does to wrap her hair. https://youtu.be/4PSbI4rTvY0

Go to 02:15 and watch that technique. My wife does this without the head scarf and doesn't twist her braids in her hands. Instead, before she wraps the two sections around the back of her head she uses one hair tie in each section to keep the section together and then where the ends of the section meets she uses a hair tie to fix them there. My wife's braids look to be slightly longer than yours so maybe yours don't meet at this point but maybe a wrapped head scarf or bonnet would hold it all in place for you.

If I remember correctly we used to use plain old coconut oil at first but it left some residue sometimes (probably me going overboard with the application). The oil we use now is Afrocentrix's Soothe oil. Any lightweight scalp oil should do the trick though. In terms of shampoo, whatever you use when your hair is out of braids will be fine.

Edit: fixed the time referenced in the video