Background images are not limited in size. You can use any size. However, you might want to use image larger than FullHD image (1920×1080) so other users with larger display resolution than Full HD don’t see your background color on the sides.
If you are using a 1024×1024 image it will look perfect for displays up to 1024px in width but for larger displays it will look like crap.
It all depends on your main target audience (check Google analytics screen resolution data) to understand what image size would fit the best.
There are a lot of options for background in this theme just play around with them to get the result you want.
For social buttons you can use any plugin that does that like http://wordpress.org/plugins/floating-social-media-icon/ or any other. You like or even your own icons which is even better since you will have full control over the looks.
Social icons soon to be included in the default theme installation. I am already working on it but can’t give you exact release date just yet.
Social sharing buttons used in the theme demo are done by using plugin called Jetpack.