It could work except the weeds will grow up between the vines and you won't be able to cultivate the corn row. If you're okay with weeds there or you love to pull weeds, go for it.

It's better to grow corn in blocks or wide rows since it's wind pollinated. If you plant it in a single-file row, the pollen has a better chance of just blowing off to the side and not onto the silks. Don't worry about sun ~ the corn will get plenty (remember, you're in Texas ;). If you still want to grow it in a line, I'd hand pollinate to make sure you get big, fat ears.

Weeds will grow up between the melon leaves, but not as many as without them. I plan to try something similar this year (squash around the base of tomatoes) planting it much thicker than the recommended spacing. I've done this before and it shaded almost all the weeds. The squash yield per plant wasn't as much, but the yield per area of ground was bigger.