Since consuming them with the Death God is how the 3rd beat the two he was fighting, I'm assuming that no, once the Death God eats your soul, you can't be brought back as a zombie.
But Madara didn't use the 1st DNA to Edo Tensei him. He used it to grow Zetsu's.
Orochimaru is the one that Edo Tensei'd the 1st. And he never fought Hashirama for his DNA that we know of.
I guess it's a question of, did Madara give Orochimaru the 1st's DNA, or did Oro get his own sample. Since we know he used it to make Yamato and to infuse Danzou with the 1sts cells.