So, for instance, the rabbis would sometimes count hei (ה) as if it were chet (ח) (as in, yShabbat 7:2). Are there other letters that you are allowed to switch when computing a gematria or deriving a notarikon? Do we ever get an explanation of why these letters can be exchanged for one another?
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There are many different substitution systems. I'm not aware of any place that puts them all nicely in a catalog, but in Sefer HaErchin Chabad in the section (two different volumes of it, actually) about letters it intersperses many examples of the types of substitutions.
The one you are looking at is based on the part of the mouth where the letter is formed (throat, pallet, teeth, lips or tongue). אחה"ע are all in the throat, and thus substitutable, but I think there is a preference for the outside letters of that order to substitute each other, and the inside letter to substitute each other (Similarly בומ"ף are a group associated with the lips, גיכ"ק with the pallet, זסצר"ש the teeth and דטלנ"ת from tongue - Sefer Yitzerah 2:3).
There can be variations on such substitutions, like the א with the ג where one is substitution the first of one group for the first of a different group, and similar ideas.
Another popular one is א"ת ב"ש - where the first letter substitutes for the last letter, second to first to the second to last, etc.
A third kind is אטב"ח - it is the same principle as the previous one, but letters substitute within their grouping of 1s 10s and 100s. So Aleph with Tes, Beis with Ches and so on, then Kof with Tzadi, Lamed with Peh, etc. In that system the middle letter (ה, נ, etc.) has no substitution.
In that Sefer HaErchin it explains at length according to Kabbalah and Chassidus (mostly Kabbalah) what many of these substitutions mean.