In your example, might indicates what is or was expected.
Still, dislike him as they might, Abu-'Ali's neighbours and kinsmen
also held him in fear. The children of the hamlet were always careful
to be discreet when they mimicked him: they would look up and down the
lanes to make sure that neither he nor his burly eldest son, 'Ali,
were in sight...
This might is confusing because it brings to mind the idiom try as one might, which means to make a great effort but still be unable to do something.
Wouldn't it be better to say instead disliked him as they might have
because the writer is talking about the past here?
No, not at all. That structure would be ungrammatical. Furthermore, might have can indicate the non-occurrence of an action or state.
If Mike had fallen, he might have gotten hurt.
Also, shouldn't it be was there instead of were?
You are right. From the book on English grammar that I have written:
Neither/nor is used to join two negative ideas. For example, Neither
the players nor the coach is wrong about which days are off.