Tiny Hut states that it:
remains stationary for the duration
If it is stationary relative to the surface it was cast on (as long as the surface is at least 20' in diameter), does that mean it's possible to cast it on a moving surface like a vehicle (boat/carriage/iceberg/etc?)
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.
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.
Leomund Tiny Hut is stationary relative to the caster position at the time of the casting.
Range: Self (10-foot radius hemisphere)
- A 10-foot-radius immobile dome of force springs into exisence around ad above you and remains stationary for the duration.
The hemisphere has a floor. Apparently hemispheres have floors, domes don't. The hemisphere word in range prevails over the dome word in the description text. as this tweet by Mr. Crawford states.
Since it never mentions the ground, we can consider RAW:
Finally: Assuming arrows were inside the area when the spell was cast, they can be shot outside. Free sniping.
Creatures and objects within the dome when youc ast this spell can move through it freely.
Treating it like Wall of Stone eliminates the million different Hover House scenarios. There's just too many ways for relatively low level, relatively clever players to create an almost impregnable floating fortress otherwise.
