Now on the second one, I'm doing sort of this schedule:
- mornings to lunch is work time, my wife looks after the little one, the older one is at kindergarten - lunch is not short break, usually from noon to 2pm, time with the wife and with the little one - 4pm-ish full stop as the older one gets home, I don't accept meetings after this time - 8pm-ish bed routine, sleep 1hr-2hr or so with them - wake up, grind some work until hitting a goal - weekends: I do some prep work for the week, but is 3 hours tops in 2 days
The little one is 1y old now, and this have being going like that for 6m or so, is it healthy? No, you need proper sleep time to be able to give care.
Do I work 8h day? Definitely not, 8h is really rare right now.
I've tried shifting my schedule to start very early, like waking 5-6am but I did not managed to switch.
This does not work on the long term I believe, there were weeks where I fully skipped the night shift and only did the work I could during the day, as I felt that I wasn't able to give care and attention properly being tired.
Furthermore, the kid has needs too, and one of those is to occasionally get your full attention, and for you to not be stressed out. I'm not sure my kids have fully recovered either.
I would use all the vacation, family leave, and sick time first, and then consider going into debt to get your wife through school.