I was corrected with the following: "I am not against him or against her." That is not proper English! You should allow for "I am not against him nor against her."
They should. This is a Hebrew course, not an English course, so they should be liberal in what they accept when it comes to English.
But isn't it supposed to be "I am neither against him nor against her" ?
No. There is no way to split נגדו. You can't say "נגד אותו" or "נגד הוא", so you can't avoid repeating the נגד
That is a verb, but it doesn't mean that, at least not in that form.
The verb מתנגד goes with the preposition ל to form sentences like אני לא מתנגד לו. The form you used נוגד might refer to an antidote or perhaps some chemical reaction.
So sentences like זֶה נוֹגֵד אֶת חֻקֵּי־הַטֶּ֫בַע this contradicts the laws of nature, זֶה נוֹגֵד אֶת כׇּל־מַה שֶּׁאֲנ֫חְנוּ מַאֲמִינִים בּוֹ this goes against all we believe in or זֶה מַמָּשׁ נוֹגֵד אֶת הַפִ֫יזִיקָה שֶׁל הַיְקוּם it really defies the physics of the universe are not used any more in Modern Hebrew, but only a sentence like אֲנִי צָרִיךְ נוֹגֵד־דִּכָּאוֹן I need an anti-depressant or מִי נוֹתֵן לַבָּחוּר אֶת נוֹגֵד־הָאֶ֫רֶס who gives the guy the antivenin?
No, they are used. But you wouldn't use this form to refer to a person opposing something.