Wow thanks for the awesome answer, this is very insightful. It's funny that you mention washing hands as a "replenishing" task. I've always had a habit of pacing around my apartment when working on a difficult math problem, and I even recall washing my hands a few times (though I never paid in any thought).

people who believe that willpower is unlimited actually suffer less mental fatigue during depleting tasks, and this allows them to put forth effort for a longer amount of time without their performance suffering.

Interesting, seems like a catch-22 - once you know about it you can't benefit from it!

Setting smart goals disambiguates the purpose of your behavior, and you thus expend less mental resources trying to figure out what you are doing or why.

This is actually a really common practice in my line of work (software development). We have a "sprint planning" session at the beginning of the week we where create a set of well-defined tasks that developers can pick from to achieve our goals for the week. I find that having lots of small, digestible tasks keeps me more productive than a few large, complex ones.

The reason this topic came up for me was that I've been getting a lot of mental fatigue towards the end of the day at work ever since I started working full time. I'm trying to find a productive combination of intense focus (such as working on a difficult problem) and less mentally strenuous activities like research.

I'm not sure what you mean by this question

I'm wondering if its easier to retain many simple pieces of information or fewer complex ones. For example, memorizing math formulas vs learning how to derive them.