하 + 여 -> 해 (하여), but most other verbs (in fact, all of the ones I know about, if I'm not mistaken) take 아/어. Is there any other verb that conjugates with 여?
No, 하다 is the only verb that conjugates with 여. The reason is a bit grammatical.
Think about regular and irregular verbs. Korean, just like English, has those 'irregular verbs' as well.
When you change the form of a verb or an adjective, it is called 용언의 활용(conjugation of declinable word). If it changes regularly, then it is called 규칙 활용(regular conjugation). If it doesn't, then it is called 불규칙 활용(irregular conjugation).
Many verbs conjugate with -아 or -어. These conjugations are 규칙 활용;
막다 + -아 : 막아
먹다 + -어 : 먹어
However, there are some verbs that conjugate irregularly, which is the case for 하다. It is called '-여 불규칙 활용'.
Of course, other irregular conjugations also exist. To list a few:
-우 불규칙 활용 푸다(ladle) + 어 : 푸어(X), 퍼(O)
In this case, the vowel ㅜ dropped out.
르 불규칙 활용 흐르다(flow) + 어 : 흐르어(X), 흘러(O)
The vowel ㅡ in 르 drops out, and ㄹ in 르 becomes ㄹㄹ, Making 흐 ㄹㄹ 어 -> 흘러.