The outline can be very hit or miss. Most people choose to add several rubber bands to guarantee the outline, and don't mind it thick. If you want one that's thin, your best bet would be to dye normally then trace around the Mickey head with a bleach pen (must be bleach, stain pens won't work). Should you choose this route, make sure you put something in between the layers of the shirt so you don't bleed through.
As for those solid color shirts, they're done as bulls eyes. In non Mickey head ones, you pick up the fabric (both layers) from wherever you want the middle of the bulls eye to be, and lift straight up. The easiest way I can think of to describe it is imagine taking a fabric napkin that's laid flat out, pinching it in the middle and picking it straight up, then pulling it through pulling it through your hand with your first finger and thumb together, like you were smoothing it into a tube. You'd band down the "tube". I'm sure you could google or youtube it and see what I mean. Then you just put on the dye. The white circles are from the rubber bands.