A simple way to look at it is that to use 「後に」, a word or phrase must precede it. Consider these examples: 後[b]で[/b]メールする。 　　(NOT 後にメールする） [i]I will email you later.[/i] 仕事した後[b]に[/b]メールする。 [i]I will email you after I've done the work.[/i] Here is its use with a noun: そうすけの後[b]に[/b]つづいてあるく。 [i]I will follow after Sousuke.[/i] Remember that the particle 「に」 can have the effect of narrowing the time window of a occurrence to a specific point, such as the difference between 「～とき」　and　「ときに」. It works similarly with 「後」 - when you use 「に」, you aren't referring to a general time frame of "sometime later," but rather to a precise moment in time. That's why you can say: 「また後[b]で[/b]やる。」 [i]I'll do it again later.[/i] but not: 「また後[b]に[/b]やる。」 [u]incorrect particle[/u] This statement isn't specifying a particular time to do it, just a broad time frame of "later." Therefore, 「で」 is used rather than 「に」。

When a verb is used, the casual past tense is required, so you would not be able to say たべる.

Not with あと. You will sometimes see [b]のち[/b]に, which is another reading of the same kanji, or ごに (in compounds, after nouns, e.g. 十年後に), but it's best just to memorize あとで and まえに as sets.