Whether pulling or pounding nails the foremost concern is leverage - this means grip the handle as far from the head as possible.
Thumbs up, thumbs down depends on the where the nail is in relation to your body and how you can best achieve maximum mechanical advantage (least effort to get the job done). Personally, I think that if I were dismantling a pallet, I would lean the boards containing the nails against a table, pound them through as you indicated, then seat the claw so that I could pull the handle downward such that body weight could help provide some of the force.
Always seat the nail as far into the space between claws as you can - keeps the hammer from slipping off and increases mechanical advantage. If the nail is too far out to get a good grip near its head, place a block of wood against the nail and then place the claw on the shaft of the nail. See below which is also a slick wayy to ease the damage done to the board.
Of course, one of the big problems with removing nails is gaining a purchase on the nail head. If you can drive the nail out from the other side far enough to attach the claw, then no problem. But what do you do when you have a 16 penny sinker holding together a couple of two by fours? Almost impossible to do without messing up the wood. First see whether the nail head is totally sunk, if not, with little effort you might be able to get the claw to get a grip. If the nail has been driven home (or beyond) your best bet is find a way to pry the boards apart (think crow bar, pry bar, or even the claw of your hammer). If these don't do the job, then you resort to solutions that are beyond the scope of this question and perhaps even for another site.