I'm trying to play KQ7 but it won't run in full screen. When I first try to run it starts but then minimizes and I have to click on it in the task bar to get it back up again. It's then stuck as a small window, if I try to alt-enter it just minimizes again.

I downloaded the KQ7 & KQ8 games from here and never had that problem. So I will list a few things I'd try if this had happened to me..

1st, go to Program Files\GOG.com\KQ78\KQ7 and open dosboxKQ7.conf with notepad.
Change the line that says output= to overlay. So that it reads "output=overlay". Make sure fullscreen is set to true.

Then right click on Sierra.exe and click on Properties. Find the Program tab and make sure Run: is either set to Normal window or Maximized. (I'm running Win7 so that's where it's located for me)

I dunno if any of that will work since i've never had that problem. It doesn't sound like a problem with the game itself, but more of what your OS is doing to the window when you try to run it