What are the best dimension for the background fullscreen images?.
There is no default or ideal size for images as fullscreen background, It all depends on what you want.
If you want better quality you have to use bigger images, like 1500x840px or more. But you will get a page to load slower, because the images are heavier in MB.
Also you can use smaller images, like 900x500px and get faster load times in your site.
About proportions, the best thing here is to go for a 16:9 ratio. This is the proportion in Full HD monitors, but the images are going to be cropped anyway because the Theme adjusts them to the size of the window.
But you can choose any proportion and see how it fits your design. It’s a matter of try and see what works best on your site.
You can not avoid the image being cut because the Browser window or the screen can be any size, and some users will see your site in a Desktop computer, other in a laptop, etc.
But with a 16:9 proportion I think you are safe in most cases.
INFO This subject was also discussed in the support forum : image sizing.