The wording can be seen to be ambiguous.
The spell description does not mention the spell being confined to earth (the element). A fair interpretation or ruling would be that you strike "whatever you are standing upon" but that leaves open some interpretations such as those you present in your question.
PHB p. 231: You strike the ground, creating a burst of divine energy
that ripples outward from you.
(Constitution save for half of 5d6 radiant plus 5d6 necrotic or thunder as chosen by the cleric)
Since one could interpret this as "ground" meaning only earth or that which is attached to the earth (how would the wooden floor of a second story building apply?), and since some folks are rules lawyers and/or very literalist (word over spirit of the rules) in their take on rules ... it would be a good idea to discuss this detail with your DM before you begin the adventure so that you know where it does and does not work. This discussion will clear the air and help the DM decide where it works, or doesn't, ahead of time.
How would I rule it?
If I am in a building on the land, and on the second floor, and that has a wooden floor, I'd still be able to "strike the ground" and make the spell work. Likewise on the deck of a ship, or on an iceberg.
KISS principle in action.
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
That's up to the DM.
As usual, there's no single clear answer to anything that isn't explicitly stated in the rules. A DM could certainly decide that waves represent an uncertain surface that the PCs will have to make rolls to move across; but they could easily rule the other way, since it's magic that says you can move across water 'as if it were solid ground', and solid ground is not generally known for heaving up and down under your feet. The latter interpretation does have some basis in our world; the original Biblical example of walking on water, which presumably inspired the spell and ring, took place in a serious storm with large waves.
What's your goal?
In general, I think the real answer comes from answering the deeper question, "What do you want to accomplish by calling for rolls?"
If there's a fight or other challenge happening and you, as the DM, want the waves to count as an environmental problem that impacts the PCs but not their aquatic enemies (thus increasing the difficulty level of the encounter), then I think that makes a pretty great fantastical setting for the scenario.
By contrast, if you're considering just having the PCs roll some checks to cross the stormy area, but those checks don't come with any actual consequences for failure (usually taking a longer time to cross an area and looking like an idiot while doing it aren't actually consequences), probably just skip it and move on to the next point of interest. You can describe them stumbling and sliding across the waves if you want to have a comedy beat, I suppose, but calling for checks in this scenario sounds a lot like the classic newbie-DM mistake of having the players make tons and tons of inconsequential rolls.
It's also worth asking yourself if your plan is eliminating the benefit of the magic item in question. If the ring is allowing the player to walk on water, but you're making them functionally perform the same rolls you'd call for from a swimming character, then you're kind of taking away the coolness and benefit of having a magic item that's perfectly suited to this challenge, and that's usually a bad thing.