You're overcomplicating things. Just set a 10-second looped timer and go for it.
There's no "stop" (until you get to the end of the whole set).
Clap push-ups are an explosive exercise. They happen very quickly.
10 seconds jacks
push-up
10 seconds jacks
push-up
10 seconds jacks
push-up
10 seconds walk-outs
push-up
etc.
There's no stopping.
The push-ups are not instantaneous, true. So they do take up a small portion of each 10-second interval. Don't worry about that.
One set of this workout takes exactly 3 minutes to complete. During that time you will execute a total of 18 clap push-ups, plus do all the other listed exercises.
You could add in extra time for the push-ups if you wanted to. Just set the timer loop for 11 seconds instead of 10. I don't think that's necessary. The workout is plenty challenging enough at 3 minutes per set! But it's up to you.
Whatever interval you use, however, don't think about the push-ups as "stops". You transition seemlessly from exercise to exercise, without stopping. It's 3 minutes (or 3:18, if you want to keep the push-ups extraneous to, instead of included in, the 30 seconds for each exercise) of intense work, followed by a 2-minute break. Rinse and repeat 3, 5, or 7 times in total.