First "He ignoring her" is not correct in any position. The subject of a gerund does not use a subject pronoun.
"She hates him ignoring her" and "She hates his ignoring her" are very close in meaning. In formal writing, the latter is preferred. In less formal contexts the former is common.
There can be a slight difference in meaning, the first suggests "She hates him when he ignores her" the second suggests "She hates the act of ignoring, when he ignores her."
You can often re-phrase, if using "his ignoring" seems to old fashioned: "She hates it when he ignores her." is correct and idiomatic.
The same goes for the gerund in subject position: both "his" and "him" are correct, but "his" is more formal, and you can rephrase "When he ignores her, she gets annoyed."