No, there are no weapon size rules in 5e beyond Heavy weapons being off limits to size Small races.
There is no feature like that in the Elemental Evil Players Companion entry for the Goliath race. Powerful Build only applies to carrying capacity and the weight limits for moving things around (push, drag, or lift).
There are no rules for weapon size in general in 5e, thus there are no size large/size giant weapon rules with wielding penalties and damage bonuses as in previous editions. PCs wield weapons appropriate for their size with no restrictions except Small creatures being unable to wield heavy weapons.
Size
Characters of most races are Medium, a size category including creatures that are roughly 4 to 8 feet tall. Members of a few races are Small (between 2 and 4 feet tall), which means that certain rules of the game affect them differently. The most important of these rules is that Small characters have trouble wielding heavy weapons, as explained in chapter 6. - (PHB p. 17)
Heavy
Small creatures have disadvantage on attack rolls with heavy weapons. A heavy weapon’s size and bulk make it too large for a Small creature to use effectively. - (PHB p. 147)
Goliaths already have some fairly strong racial features and do not need a permanent racial damage buff.
No, it's not.
When lifting the other character you are either taking an action, or you interacting with an object -- depending on your DM's rulings.
If the DM rules that the pile driver is an attack (which I'd argue it is), that would cost you an action, and you can only repeat it if you have movement and an action to spend on it. This assumes you've grappled them, and have an attack left.
If the DM rules you are use an object. If it wasn't an enemy, you'd only be allowed to pick it up once. Once you drop it (which is free), it is no longer a free action to pick it up again. Use An Object action:
You normally interact with an object while doing something else, such
as when you draw a sword as part of an attack. When an object requires
your action for its use, you take the Use an Object action. This
action is also useful when you want to interact with more than one
object on your turn. (PHB 193)
Since you can't lift someone and do something else, it might take a full action to lift someone above your head. Remember that a round is 6 seconds, dropping someone more than once (or twice if you have the actions for it) would take about that long. Moreover, really can't lift someone and jump while doing something else, so it could be argued it can't be an interaction and must be a "Use Object" action.
To rule otherwise would allow a free 5d6 damage without using any action, and that, while awesome the first time (and a DM may allow it once, by Rule of Cool), is game breaking.
Best Answer
As RAW, no, it does not
Being considered a size higher for carrying capacity and push, drag, or lift force is not the same as being a size higher for all purposes, including grappling.
As RAI, maybe
The loop hole is that the PHB has no reference on why and how the grapple-er can move without speed penalty if it is two size higher, and the DMG does not touch the subject. There are two potential scenarios for why a grapple-er one size or lower has it speed reduced, though; the grappled creature have enough leverage to make the movement difficult or that the grapple-er cannot carry comfortable enough the grappled creature as in "difficult terrain", and that the weight of the creature is the deterrent.
In the first case, since Powerful Build does not increase per-se the size, it is safe to assume that it does not help to increase the comfort or reduce the leverage and, such, the speed is halved.
In the second case, where is the weight that maters, things change. Powerful Build improves the carrying, dragging, and lift capacity, therefore the grapple-er creature does not have its speed halved.
Which one it is
The only evidence I found is in favor of is that weight is the reason for the reduction in speed. The first part is in the text of moving a grappled creature (emphasis mine).
As far as I can tell, there are no rules on how you drag or, particularly, carry a grappled creature. You can carry over you head a grapple creature as if it were a sack of potatoes. In PHB 176 there is a segment that shows how can you drag, carry and lift something in particular and, as shown, it is a STR and weight contest in which Powerful Build should work as intended.
Bottom line
At the end is the DM that has the last word. Talk to your DM and present the evidence. If you are the DM decide what it is best for your campaign. If a particular decision is more powerful that you expected it is in your right to take back the decision.