I hate it when he does that.
I hate when he does that.
This is very common (22.8 million Google hits for “I hate when” this morning). It is informal and maybe not completely standard English.
As you note, whether it is present or dropped does not affect the meaning.
The verb hate in this case is still transitive. The phrase when he does that serves as the direct object of hate.
This quirk isn’t specific to the verb hate. There are 21.4 million hits for “I love when”, and I found these too:
Like any small boy, he resents when his mother shows him too much affection in public...
While it’s great to see any old horror movie projected in glorious 35mm, I really dig when it’s a movie that’s never been released on DVD.
I like when my scissors glide through the paper so I don't have to cut.
Is it related that I find the phrase "write me" or "I'll write you" odd?
No, this is standard, and it seems more verb-specific to me than the I hate when construction. Oxford Dictionaries Online gives this example:
Mother wrote me and told me about poor Simon's death.
There might be many other verbs that do this. The one that comes to mind right now is tell:
She told me a long story. (recipient is indirect object; story is direct object)
She told a long story. (story is direct object; recipient omitted)
She told me. (recipient is indirect object; story omitted)
Clearly many verbs that can take an indirect object don’t work this way (give, get, read, hand, bake).