oil disappearing from pan
Simply put, your oil isn't hot enough. If you add vegetables to cold oil in a cold pan and then heat everything up, your veggies (or just about anything else) will soak up the oil. Heat the oil in the pan until it starts to shimmer and a drop of water sizzles, dances and disappears. THEN add your veggies.  The very definition of "saute" comes from "to jump and to dance". It's all about the hot oil. Be wary of your oil's smoke point, don't let it burn, but get it good and hot. 

EDIT: A friend in chat dug up this video. I like that it shows using water to test the heat of the oil. If you're trying to brown slices of eggplant, the way she treats shrimp is very similar to how you want to handle eggplant slices. Stir-frying is a different technique, but the hot-oil rule remains the same. Saute Video