Yes! Cabbages and humans share about 40-50% of their DNA. Humans even share some DNA with microscopic yeast! This is because all life on this planet has a common ancestor, that means that life is only known to have originated once and all stems from that!
Bacteria, Cabbages, even Elephants all have the same common ancestor if you go back in time far enough and there is a surprising amount of DNA, such as that between cabbages and humans, that remains little changed since they diverged from their common ancestor.
(Also, about 50% of human DNA is shared with bananas!)
Yes they do! I was curious about what genes they share, but couldn’t find anything overly exciting – the only one I could find is cytochrome c, which is a protein found in mitochondira. (http://en.wikipedia.org/wiki/Cytochrome_c)
Nature is lazy – if something works, it will probably stay. My guess is that a lot of the genes/proteins that are involved in the basic stuff (such as reading DNA, generating energy and transporting ions) would use very similar between human and cabbages.