(C) is the only possible answer to the question.
(A) and (E) both contain the unsupported pronoun "they". Every pronoun on the GMAT will have an antecedent (the only exception to this rule is "it", which can appear in reference to the universe as a whole, e.g. "it is a nice day today").
We can also eliminate (B) for verb tense problems. "as" indicates that things are going on simultaneously, so switching to "had just been" is wrong.
(D) has modification issues. We need to remember that the part of the sentence after the underlined portion also has to make sense. Looking at the fragment:
"during the arrangement of a bouquet of roses just picked from his garden in a vase on his bedroom windowsill"
we see that (D) makes is sound as though the garden is in a vase on his bedroom windowsill, which makes no sense.
Only (C) gets everything in the sentence where it should be, uses the correct tenses and doesn't introduce any ambiguous pronouns.
As an aside, it's certainly correct that "being" is rarely correct on the GMAT. A good general rule of thumb is "other things being equal, don't pick choices with "being" in them". However, avoiding "being" is a style choice, so if the only gramatically correct choice uses the word, that choice will be correct.