How bad can the greedy coloring (list color) for the c-chromatic number of graph be?
nice question.
Consider the following construction: build k P3s numbered / ordered as 1-2-3 4-5-6 7-8-9, etc.
Now 1-2-3 all get colour R by the greedy scheme. Make 4,5,6 all adjacent to 3. Then 4,5,6 each get colour B. Now make vertices 7,8,9 adjacent to 3 and to 6, then they can't get colours R or B. They get Y.

Continue with 10-11-12, with 10,11,12 adjacent to 3,6,9. They cannot be coloured with R B Y, so they get G.

These 3k vertices will need k = n/3 colours. But note that c(G) is 2, since the set {3,6,9,12,etc} induce a clique, and the rest of the graph induces a perfect matching. So this shows that greedy colouring can still be arbitrarily bad for cograph colouring as well.

We can modify this construction to cut down the max degree as well... 4,5,6 are adjacent to 3, but make 7,8,9 still adjacent to 6 but now adjacent to 1 instead of 3. Then make 10,11,12 adjacent to 9, 4(instead of 6), and 3. And for future triplets, keep alternating which end-vertex of a prior P3 they join to. The resulting graph is probably not partitionable into 2 cographs anymore, but observe that 2,5,8,11,etc form an independent set and the rest is probably coverable with 2 more cographs, but I'm not quite sure. But I don't think this is important ... what is important here is that the max degree is low enough that the greedy colouring uses more than $\frac{1+\Delta}{2}$ colours (to answer your second question.)

Another interesting question to ask would be whether a "smarter" greedy colouring (such as LexBFS) would produce a constant-ratio approximation.