In Catholic doctrine, are plants living beings?

No. Only things that have what the Bible calls the breath of life are living beings in that sense. In Noah's flood, Scripture says all that had the breath of life died, and that does not include plants; God breathes the breath of life into animals and men, and that is the distinguishing feature. Plants are of course organic, carbon-based entities like any tree, fish, or bird, so we share elemental properties with them, but they are not living beings in the way animals and humans are, which are the ones considered to have the breath of life.