Instead of writing: "After he had done this, he did that." Is it acceptable to go: "After he did this, he did that." That sentence sounds reasonable. However, I don't understand why. Since I'm claiming that he had already done "this", why isn't the "had" necessary? I could use some explanation for this. Thanks so much.
can't answer it generally, like that... give us some real examples, in context, and we can give you valid advice...
If it's the same person why not, "After doing this, he did that"? Instead of After she took three long strides, the bat woman leaped into the air and flapped her massive wings. I don't see a problem with either one. After he pounded on the door for five minutes, he gave up. In most cases, if it is the same person in both clauses, I use the "After doing this, he did that" structure.
Yes, generally, if the order of past events is clear (as it is here made clear by 'after'), there is no call for the past perfect. Occasionally it is used for emphasis, and to make clear that the earlier event was completed before the onset of the subsequent event.
As I see it, "after" and "had" serve the same purpose in that sentence, (to explain "this" happened before "that") so the "had" is redundant and that's why you can omit it. You'd only have a problem (maybe) if you wrote: "He did this. He did that." That could be potentially confusing, but even then, events are generally assumed to be sequential unless we're told otherwise. In fact, I usually write that way. . .
Maia is correct in that you have given only a generic example of two different syntax constructions without the context that would validate or invalidate the use of either one or the other.
grazie, wrey!... seems a waste of time to speculate, without context... but starting the sentence with 'after' isn't usually a good idea, in any circumstance, imo...