To save you a trip to the dictionary -- The 'sheets' here (arkene) refer to PAPER, not bedding.
Nothing in this case. I can't think of a situation where they couldn't be used fuldstændigt interchangeably.
Ethymologically it looks much closer to "completely" if I assume the similarity with German... fuld-stændigt sounds the same as vollständig (which is also broken into pieces voll-ständig, and also means completely).
Why isn't this "fuldstændige" since the other words are in their plural form?
In this sentence, "Fuldstændigt" is an adverb modifying the adjective "hvide." Typically, adjectives agree with the nouns they modify, but adverbs are invariable. In this example, the ending -t on "fuldstændigt" helps you to recognize the word as an adverb. Does that help?