There is a whole class of such nouns that exhibit vowel fronting when used as standalone nouns, and not all of them end in -e or -wa. Among other examples, with in-compound forms followed by standalone:
- 神: かむ vs. かみ
- 天: あま vs. あめ (also for 雨)
- 口: くつ vs. くち
- 目: ま vs. め
- 手: た vs. て
- 月: つく vs. つき
- 木: こ vs. き
One of the reasons for the theory that a certain class of nouns was followed by the now-obsolete Old Japanese い, an emphatic nominalizing particle, is that the term カムイ appears in Ainu as a likely borrowing from Old Japanese or slightly earlier, clearly manifesting a distinct む and a distinct い sound. These two over time could conceivably mush together into み, much as similar sound shifts have been observed in even modern Japanese (such as たかい becoming たけえ, すごい becoming すげえ, etc.) and in other languages around the world.
As to why only certain nouns evince this particular phenomenon, it is not unknown in other languages for there to be specific noun classes. It's possible that these nouns in Japanese might be a vestige of an earlier stage of the language that had such a specific noun class. Notably, a lot of these nouns (at least, the ones I'm aware of) seem to describe parts of the body, spirits, and other concepts that would be personally important. Polynesian languages have a roughly analogous noun class covering so-called inseparables, and these nouns take a specific version of the possessive particle, "no", contrasting with the "na" particle used for possessed nouns outside of this class.
Regarding transitivity / intransitivity and verb conjugation patterns, see blutorange's extensive post about this subject. There are several patterns, of which -わる / -える is just one.