Does acrolein give Fehling's test? Aromatic aldehydes do not give Fehling's test, which makes me believe that for the same reason acrolein should not. It has a double bond conjugated with the carbon-oxygen double bond as in benzaldehyde. Is my assumption correct? Or is extended conjugation a factor?

Glyoxal(CHO-CHO) and Glyoxalic acid(CHO-COOH) also don't give Fehling's test. Any reasons for that?
It has nothing to do with aromaticity. It is related to this answer.

Why does benzaldehyde not respond to Fehling's test?

Any aldehyde without an alpha hydrogen will not give Fehling's test.
Acrolein also doesn't have an alpha hydrogen, so it will not give Fehling's test.