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The third perek focuses on three final areas:
(1) Mishnayot 1 and 2: In general, the Sages desired to push us to begin Yom Tov with a clean and comely body, and with a clean wardrobe. One tool they deployed to this end is to prohibit haircutting, shaving, and washing clothing during Chol HaMoed, thus compelling us to do these things before Yom Tov starts However, the Mishna recognizes a variety of circumstances in which a person may have been unable to perform these activities before Yom Tov due to circumstances beyond their control, or in which having done so before Yom Tov does not obviate the need to do so again on Chol HaMoed (e.g. clothing that will invariably become dirty and will be needed multiple times over the course of the holiday).
By the way, much has been written about the applicability of the shaving prohibition in our day, see for example here . And with regard to washing clothing, there are some contemporary leniencies that are built upon the fact that washing clothing today requires so much less physical effort than it did in Mishnaic times.
(2) Mishnayot 3 and 4 limit the situations in which writing may be done to situations in which the failure to write will likely result in financial or other loss. The laws around writing on Chol HaMoed also became more lenient over time, as writing moved from being the professional work of a scribe to the activity of anybody with a ball point pen.
(3) The final Mishnayot deal with the collision of Yom Tov / Chol HaMoed and personal mourning. The practical Halacha is often at variance with what is stated in the Mishna. For example, it is our practice that if a person began sitting shiva even for an hour prior to the onset of Yom Tov, that Yom Tov cancels the rest of shiva. Also, we do include Rosh HaShana and Yom Kippur in the category of "Yom Tov" for these purposes.
A final note: I remember an article written by Rabbi David Hartman in which he disagreed with his mentor, Rabbi Soloveitchik, about the underlying ethic of Yom Tov suspending shiva. Whereas the latter thought it was about Halacha's demand that a person suppress his personal emotional pain in order to join the public state of joy, Rabbi Hartman believed that the Halacha never asks us to suppress our emotions. What is being asked here rather is that we not detract from the community's joy through our exhibiting outward mourning rituals.
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