Nothings wrong with your car the coolant water spills out of your car when its boiling hot inside which happens when you are going up the steep slopes, you will notice a drain area on the bottle cap. 1000cc with auto gear is very hard to go up. I have had similar experience with my 2004 Vitz. all you need to do is fill up the radiator with water frequently in those situations. If you are not satisfied just go to a workshop and get it checked thats what i did and all was fine.
The red gear box problem might be related to gear box itself. The gear oil needs to be changed every 35000/40000 KM, have you done that? If not do it ASAP. Get it checked from a good workshop, in lahore i goto mercanics (mercedes workshop) near airport but you can goto toyota workshops and get your gear oil checked if it needs replacement. Do this to avoid larger expense of changing the whole gear box.