Will the logo I submit for branding set-up automatically apply across all my pages?
No, not yet.
You can choose which logo you want to place as you create your navigations/menu and footers on your pages. But, since navs and footers are generally global modules, it will apply across all pages if you want them to.
For example, you choose the SR Nav Dropdown 01, you can choose the logo you wish to have there:
(For more information, please see Changing Nav Settings https://docs.sprocketrocket.co/changing-nav-settings)