Syrena, On the burn, you'll need to post back with more info; first you say this is on the counter, then you say the burn is on the sink. Need to know which it is, and what kind of counter or sink.
As to the others;
- Removing old oil: Using a putty knife and rags, wipe up all the excess oil you can. Pour some automotive transmission fluid on the oil (I know, this sounds like adding to the problem, but transmission fluid is high-detergent, and will help break it loose). Using a scrub brush, scrub the transmission fluid over the oil spot, then wipe this off. Next, scrub the area again with a strong mixture of water & laundry detergent, like Tide. Wipe up the excess & rinse. Repeat these steps as needed. You can also try driveway cleaning products available from hardware stores.
- Repairing chip in bathtub. From a hardware store, get some porcelain chip repair; ask a clerk to help find it if needed. Usually this is a small bottle of material you will apply in a similar fashion to nail polish. Follow the directions carefully for prep & application. The best finish is obtained by using multiple layers, placing the repair material inside the chip area & working it to the sides, but not brushing it over the edges of the chip onto the surrounding, undamaged area; let it flow to level on it's own & dry before applying another coat.