I'll tell you want I do with sites I design. Basically I know of no one that uses a web browser full screen on wide monitors. My monitor is 2560px wide, and I never set the browser window full screen. I have a very large monitor for photo editing and also so that I can have more than one window in a couple programs open at once, side by side.
If you notice on the theme you are using, it has a maximum width for the content area of 1000px. That is so the content area doesn't get too wide to make reading difficult (typically text areas should be around 500-600px wide. The 1000px was very likely chosen to allow people using 1024px wide monitors to not have to scroll horizontally, and there are still about 20% of the webbing public on 1024px wide resolutions.
I use background images 1500px wide typically in web designs since that covers the majority of monitors out there and then I choose a complimentary background color so that if someone is viewing the site with a browser window set wider than 1500px, they will not see the stark white to the right and left of the image. For image height, I generally don't worry about going more than 1000 and then set the image to not scroll (fixed).
Images that are 1900px wide or more cannot be optimized down to a file size that does NOT adversely impact page loading times (typically they will be 1MB or more), and since the search engines consider page loading times when determining search engine ranking, slow loading sites will get a hit and appear lower in search results.
In general, try and keep background images under 150k in file size if at all possible. That will also make those of us on slow(er) internet connections happy.