Lighttpd should be fine.
I meant have the local display run its own browser, viewing stuff served by the embedded Linux. I.e. the local display would be a small tablet. Running a browser on the embedded Linux is another idea, hmm, might be ok if the linux distro supports it. Do you have an SD card slot on the board? A browser is pretty heavyweight.
No. You'll have to run something like X on the local display if you want to program it with Qt though. Is this a consumer product where you need a really slick interface, or is it something industrial where it's more about functionality?
Yes, that is called "kiosk mode" and there are usually ways to do it.