Verbs: Using "had" + Verb, vs. not. What's the difference?

Solution 1:

Americans have no problems with any of those sentences, and prefer (of course) the second, a simpler one.

British people feel the difference, and use the first one, which is in the form of the past perfect simple tense, because the whole sentence is about the past, and one action (folding) was finished in the past before the other (returning, also in the past) was started.

To your second question, "...the plane Bill had folded..." vs. "the plane that Bill had folded...", both forms are correct but the first one is more often used because of its simplicity. The second one is used to emphasize who had folded it (Bill).