let me clarify that for you.
1) Yes, on day 1 you count rest time. You start the clock and do as many rounds as you can in those 20 minutes. You can stop to rest, but the clock keeps ticking.
2) No, day 2 does not determine your level progression. The idea of timing on day 2 is just to keep you motivated to do the sets with as little rest as possible. A day 2 workout log looks like this:
Rounds: 00:52, 00: 51, 00:55, 00: 48, 00:57. Each round is timed independently, and rest time between sets is not counted. Timing on day 2 is not very important, and some people choose not to time it at all. Timing is only important on days 1 and 3.
3) On day 3, you have 5 minutes total time to complete the round. So you do count rest time. As on day 1, the clock never stops! You can break a set in smaller sets, say, 10 pullups = you do 4, then rest 10 seconds, then do another 3, rest 15 seconds, do another 4. Then you move to pushups. Say, 12 pushups total = you do 6, rest a little, do another 6. Then move on to squats, etc.
The idea is to complete all the round in less than 5 minutes. The clock never stops, so if you rest too much, you don`t complete in less than five min. That`s what determines your progression. If you complete in less than 5 min, you move up a level. Day 3 is called "judgment day".
Staying below failure is stopping before you can no longer move and resting until you can restart. You have to do the sets in the order prescribed (pullups, pushups, squats) but you can break them up in smaller sets, as I said above. In fact, as you progress, you WILL have to break in smaller sets in order to complete. It`s very hard to crank out 30 pullups without stopping midset!
In all days, if you are too exhausted to continue, even after rest, just stop. Don`t push yourself too hard, especially in the beginning, and make sure to rest for a day between sessions. Eat well and sleep well. Congratulations for starting your training!
If you have an android phone, install the simplefit app. Look for simplefit on google play. It helps with timing, logging and instructions. I`m not sure if iPhone has an app.
Check my workout: (http://www.simplefit.org/phpbb/viewtopic.php?f=4&t=2694)