As @Frank mentions in his answer, the piles aren't stopping short of the cap since the piles are more than sufficiently embedded in the wall. This therefore means that the forces go through the topping beam, down the wall and then into the piles.
That being said, if you're going to make the wall axial-load-bearing, I don't see the purpose for raising the piles so far into the wall. Steel piles usually only need around 1-2 ft of embedment length into the cap for effective load transfer. Here's a diagram provided by a Brazilian pile supplier (Gerdau-AçoMinas):
Very few codes actually define minimal embedment lengths, but most seem to hover around 1-2 ft:
- This thesis suggests (for cylindrical steel piles) 1 ft with reinforcement or 2 ft without.
- This thesis has experimental results pointing towards 1.5 pile diameters.
- This American state survey lists multiple different state's suggestions and most are up to 2 ft (only NE, NJ, ND suggest 3 ft).
- This report concludes that embedment should be equal to two pile diameters or heights.
Now, all of these assume that the pile is embedding into an actual cap, which must be sufficiently wide in both horizontal directions. For an example, here's another diagram from the same supplier (basically, there must be 25 cm of concrete around the pile):
So, if your wall isn't thick enough to count as a standard cap, I'd guess the extended embedment length allows for the forces to be transmitted to the pile at lower stresses along the wall. No idea how to quantify that, though. Personally I'd rather just thicken the foot of the wall and leave the piles at the minimal embedment length.
The load paths are simple enough. The vertical loads obviously just go right through to the piles (I'm assuming the girders lie directly above the piles). The horizontal loads go from the girders to the bearing pads to the topping beam and then, via shear friction, go to the wall (as you mention, the (c) bars would be responsible for this) and then are transferred to the piles.
Regarding the reinforcements, it's hard to tell without dimensions and loading: are they close to the minimal reinforcement? Is the anchorage length of the (c) bars sufficient?