Easy to do. Create 4 screens, as you've outlined. For each screen, set the appropriate background and add the cells you'd like to display. Basically, get each screen looking the way you want.
Then, on the Layout tab, press the Screen Sets button. What you will need are 4 different screen sets. There is already a "Default" screen set, which you can use, and you'll need to add 3 more. I'd use the "Default" on for normal play (so you "During play" screen). Create another for Pre-Tournament, one for During Breaks, and one for After Tournament. Add the corresponding screen to each appropriate screen set. For example, add the "During play" screen to the "Default" screen set. Add the "Pre tournament" screen to the "Pre tournament" screen set. Etc. Each of the 4 screen sets will have just 1 screen in the set. Finally, you'll need to add the conditions so that the appropriate screen set is displayed at the appropriate time. Select the "Pre tournament" screen set and press the "Conditions" button. Select "Before game" for the Stage and press OK. Select the "After tournament" screen set and press the Conditions button. Select "After game" for the Stage and press OK. Select the "During breaks" screen set and press the "Conditions" button. Select "During game" for the Stage, select "Break number:" for the Level, select "=" for the operator next to it, and then enter the word any in the input, then press OK.
That's it. Because we have set pretty specific conditions, you shouldn't have to order the screens sets in any particular way. The Default screen set has no conditions, but it is always examined last, so if the conditions match any of the other screen sets, they'll be chosen. It's only when the conditions don't match all other screen sets that the Default is used. So, that should work.