You can also just drag the window to the side of the screen and “push” it to the next Space.
I don’t think that there is a reliable method to change the wallpaper of all Spaces at once. There used to be some automated workarounds for this, but I don’t think they work anymore.
Another albeit dirty workaround is to create a new folder, put your wallpaper into it (just make sure that there is only one picture in it), add it to the sidebar of the settings panel, select it, choose “Change picture” and select an interval that works for you (e.g. “when logging in” or “when waking from sleep” are straightforward choices). Now you need to apply that same setting to all of your Spaces once. If you choose “when logging in” or “when waking from sleep” then you only need to logout/login or sleep/awake after you swapped the wallpaper within the folder. All of your spaces should then have the same wallpaper.