When clauses are compressed into untensed verb forms like participles and infinitives in short phrases, information is lost, which means ambiguity increases. This is not necessarily a bad thing, but it does mean you can't always tell what the words are sposta mean, especially when they're printed instead of spoken.
In the two examples given,
- Being washed at a lower temperature, T-shirts have a longer life cycle.
- Washed at a lower temperature, T-shirts have a longer life cycle.
there are different presuppositions. This is not a matter of grammar, but rather of what you mean. (1), with being washed, presupposes that all T-shirts are always washed at a lower temperature. (2), without the auxiliary being, does not, at least not necessarily, because ambiguity; rather, its initial participial phrase would normally be interpreted as a conditional (when/if they are washed), instead of a factive (since they are washed) like (1).
As to which answer would be correct, that would depend on what the actual question was. There is no automatic syntactic transformation between tensed clauses and participles, so it's lexical semantics and pragmatics that counts. Offhand, I'd say (2) was a better match than (1), but neither conveys all the information of the original.