To the excellent answers that have already been given, I would only add that "filling in" is also what you do in a coloring book. You fill in the various shapes with various colors.

"Fill in" is more often used for fields in a form. ("Please fill in your age, sex, and ID number.") "Fill out" is usually used for an entire document. ("Please fill out the application form.") It would sound weird to say "Fill out your age" or "Fill in the application form". I have never used or heard "filled out" as a synonym for "completed". I suppose it's not grammatically incorrect according to a rule in a book, but native speakers would consider it awkward. It might be because phrasal verbs are more common in informal speech and the context of "Please find attached the ____ form" is very formal. "Completed" is not considered a formal word, so it would be fine in informal speech ("Here's the completed form you wanted.")

Hi Angela! Thanks for your great questions. To my ears as a native speaker from Canada, "fill out a form" sounds much more natural than using "fill in", although you could use either one interchangeably, as they are both correct. As for your first question, we could say that "complete" is a more formal synonym of "fill out" (at least, in this context). So while it is perfectly normal and in fact most natural to speak about "filling out a form", in a sentence like the one you wrote, which would most likely be found written in a formal or official context, "completed" would sound best.