The main difference between red and green Thai curry is that the green one is made from green peppers and the red one from red peppers. The green chillies that are used are hotter than the red ones, so the green curry will be spicier. I think for that reason the green curry is often made a little sweeter, to offset the spiciness. I don't think there are any rules for the vegetables that go into both dices, when I've ordered the dishes in restaurants they pretty much looked the same.
If I were to cook it I'd probably first stir-fry the vegetables (onion, bell pepper, baby aubergine, bamboo shoots), then add the spices (garlic, curry paste) and stir-fry for another minute to release the flavors, add the coconut milk and bring to a slow boil and then add the chicken to that so that it can braise until done, as a last step you add the fresh basil and sprinkle it with some lemon juice (as a vegetarian I would actually add fried tofu at the end of the cooking process and skip the chicken

).
I don't think the type of oil really matters, the flavors in the curry will overpower it. So the best thing to do then is to take a neutral cooking oil, I guess.
I definitely would not add carrot, celery or potatoes to the curry. You can try zucchinis.