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.

There are two different things happening here: your movement speed being halved, and spending half your movement speed. These work differently, and the order matters. The end result is that the grappler can't move-drag after standing up from prone. Here's how it works:

1. You start with your full movement speed.

   Let's use 60′ for the sake of example.

2. Standing up costs half your movement.

   Your current speed is 60′, half of which is 30′. To stand up you spend 30′ of movement. Your speed is still 60′.

3. Dragging while grappling halves your movement speed. Your movement speed this round has now been reduced to exactly how much you've already spent, so you will have 0 feet of movement to spend.

   Your current speed is 60′, but attempting to move-drag a grappled opponent changes it to 30′. You have already spent 30′ of movement and have zero feet left to spend.

The end result is that after standing up, you still have half your movement left, but as soon as you try to drag a grappled opponent you will have no movement left and remain where you are, so realistically you won't bother trying to drag that round. (You could still end the grapple and move your remaining half movement though, of course.)