I think the school competition can be divided into two groups, A and B, with four teams in each group. Cycle first, then cross. So you don't have to put off the schedule for too long.
Draw lots and write A 1, A2, A3, A4, B 1, B2, B3, B4 directly on the note. Then fold it up and put it in an opaque container.
Schedule: ("the first day" represents a time period, or it can be half a day)
Day 1: A 1-A2, A3-A4, B 1-B2, B3-B4.
The next day: A 1-A3, A2-A4, B 1-B3, B2-B4.
Day 3: A 1-A4, A2-A3, B 1-B4, B2-B3.
Day 4: First place in Group A-second place in Group B, first place in Group B-second place in Group A..
Day 5: Final
Tug of war is similar.