The game tells me that on days off, I can have my character stay home to just mellow out, but this is a lie!
Every time I click go home on a day off, the game refuses and tells me it's too early.

Given the fact that this leaves a CONSTANT increase in stress over time, what am I supposed to do to lower stress?

I recently finished it on default difficulty.
Sure have a problem to lower stress on the first run, but figured out a way in later runs.
IMO, it is best to do one activity after school, and then go to stroll in the park. It not only lower stress but also increase fitness, which allow you to handle stress better. Once you get higher fitness, you can do more activity.
Memory foam mattress sounds great at first glance, but stress affect successful rate the most, I would go for alpaca blanket asap.

Thanks for the tip.
What I ended up doing was just setting difficulty to minimal.

Although before I did that, I figured out that once you can eat lunch in the classroom, it provides just enough with my previous strategy that I was able to keep it down.
I used days off to balance out the stress vs money.

I wasn't aware that you could go home after school, instead of going to train the people/manage them. :-o
That's good to know.
Hopefully helps someone else who reads this!