If the caster falls, the spell ends once the caster is no longer in the original area.
While the linked questions have some debate about whether or not the dome moves with its surroundings, the text is clear that if the caster moves out of the original "stationary" position, then the spell will end:
The spell ends if you leave its area.
Therefore, as formulated in your question, the hut would have disappeared once the original caster was no longer in that space.
I agree with the linked answers that there is no clear definition of "stationary," which leaves the crux of your question up to a DM's judgment call. As a DM, I would have ruled that the hut does not fall because there seems to be a clear "stationary" reference frame based on your description of the fight: the rest of the environment that isn't moving. It's worth noting that the range of the spell is "self," and since the hut obviously doesn't move with the caster, the location of the hut is where the caster was at the moment the spell was cast.
While other DMs could reasonably rule otherwise, it opens up a lot of judgment calls--how big does the piece of moving ground have to be? If the caster digs up the ground and moves it around, could they move their hut? In this case, where you're on solid ground, it seems easier to say that it doesn't move at all.
Leomund's Tiny Hut has a floor
However, the scenario in your question could have only happened if the players willingly fell out of the hut, because Jeremy Crawford tweeted that the intent is that it has a floor to stand on:
Leomund's tiny hut does have a floor, Mr. Crawford (read your own book). The spell's range entry says the effect is hemispherical.
Thus, even though the ground fell out from under you, you could have still stood on the floor of the hut.
As for balance concerns, many Big Bads often have some way of dispelling magic. Leomund's Tiny Hut, a 3rd level spell, would be automatically dispelled. If the gargantuan attacker were only a physical attacker, it could have simply left and waited out the spell's duration out of range of your attacks--you can't pursue it, after all, without leaving the protection of the hut.
You Cannot Teleport Into the Tiny Hut
A key, though subtle point to this opinion is a statement on Range in the PHB (203)
Once a spell is cast, its effects aren’t limited by its range, unless the spell’s description says otherwise.
The suggestion is that during and before spell casting the spell's effects are limited by range.
Jeremy Crawford weighs in on this issue here, specifically in this tweet:
The indigo layer of prismatic wall prevents you from casting spells through it. This doesn't stop a spell with a range of self, such as misty step, but it does stop dimension door, which has a range of 500 ft.
and Leomund's Tiny Hut (PHB 254):
Spells and other magical effects can’t extend through the dome or be cast through it.
Because of it's range of 500 ft, casting Dimension Door to teleport into the Tiny Hut would involve casting the spell through the dome and cannot be done.
Best Answer
Rules As Written
There's no specific definition of "stationary" in the game. Depending on your point of view, nothing is stationary, because planets whip through the cosmos at thousands of miles per hour.
That said, I've seen it ruled both ways, because there are no hard rules either way. One of the core design tenets of 5th Edition is "rulings over rules". Some things are deliberately left to the GM's discretion.
Advice
Personally, my ruling is based on mapping: if it's big enough or elaborate enough that a battle can take place on it, then it's big enough to count as a stationary object for spells that require them (like teleportation circles onboard ships).
So, to specifically address your list:
Boat - Nope, because boats aren't big enough to be a set piece on their own. Even large boats, like say a viking style longboat, don't count. They're potentially sizable, but they don't have multiple decks, compartments with doors, stairways and ladders, etc. I wouldn't bother mapping the details of that kind of boat, like I would a proper ship. To put it another way, during an encounter, boats will move around within the lakes or rivers the encounter takes place on, but a ship is where the encounter takes place.
Carriage - Nope, for the same reason. It's going to be part of an encounter, not a setting for the entire encounter.
Iceberg - Again, assuming it's large enough... yes. It's not specifically about dimensions, though - an iceberg just big enough for the spell's area wouldn't cut it. If it isn't big enough to be a setting for an encounter, if it isn't worth the time to draw out, it's not big enough to count as stationary.
To be clear, my ruling is not based on physical dimensions, beyond needing the minimum the spell calls for (10' radius for Tiny Hut, 10' diameter for Teleportation Circle, etc). It's about the importance of the area and the worthiness to be setting for an encounter.