Most dark hair (other than pure jet black) has some natural red in the underlying color. When the processed color gets old you will get orangy or brassy tones to the bleached sections. A way to compensate is to have your colorist apply a toner every 4-6 weeks as necessary to reduce the orange. This is just normal upkeep for colored hair, in addition to using shampoo and conditioner formulated to protect your color.
Natural highlights do not vary that much from your normal hair color. That is where most people make the big mistake. Dark brown hair will not naturally have platinum highlights - ever. So you want to stay with dark caramel or another color not more than 2-3 shades lighter than your natural color. That means if your natural color is a level 2-3, you don't want to go any lighter than a level 5-6 for your highlights.
With your warm skin tone and the tendency of your hair to show orange, you want to stay with a warm color (typically called "golden' on do-it-yourself haircolor). Ash shades won't look natural or be flattering on you.
Balayage is hand-applied color to small sections of hair with the size determined by customer preference. Some stylists use a backer or foil to make sure the color doesn't get on nearby natural strands and some don't. It is a technique that allows you to start with just a few highlights and you can then add more if you like in subsequent appointments. You and your stylist can decide at your appointment exactly where you want the highlights and how much you want to begin with. It gives a really nice natural effect and doesn't look as awkward when it grows out (or if you miss an appointment).
Partial highlights are done on just on just one part of the hair, typically done with foils on the topmost layers. It will leave a more pronounced demarcation than balayage as it grows out. Upkeep is exactly the same - special shampoos and toner every 4-6 weeks.
Please let me know if you have questions.