My reading: the hut's wall only blocks targeting and line of effect for spells
The relevant text from the tiny hut spell is:
Spells and other magical effects can’t extend through the dome or be cast through it.
This wording is somewhat peculiar. Notably, rather than making a blanket statement that the hut blocks all magic, it lists two specific things that spells and magical effects can't do. And it's a good thing it doesn't say the dome blocks all magic, because if it did, then none of your party could carry any of their fancy magic equipment through the dome, which would be an unpleasant surprise for your friend wearing a Ring of Spell Storing and a Belt of Giant Strength, or your ranger carrying a quiver of +1 magic arrows on their back. Furthermore, if the hut blocked the passage of all magical effects, anyone with a spell cast on them would be unable to pass through the dome until the spell ends. Do you think it makes sense that casting guidance on a creature should trap them inside the hut? I don't.
So, by the same logic, your deadly swarm of animated caltrops should have no more trouble passing through the dome than they did before you animated them. Passing through the dome is not the same as "extending" through the dome, and the caltrops are not casting a spell through the dome, so the tiny hut spell's text about blocking magical effects doesn't do anything to stop them from passing through. The caltrops only need to be inside the hut when you cast the spell, so that you can target them.
What the dome does do is act as total cover for the purposes of spell targeting and areas of effect, in exactly the same way as a mundane dome-shaped wall would, but with the additional feature that even spells that don't require line of effect are blocked.
This interpretation avoids self-contradiction
We can also note that interpreting the magic-blocking feature of tiny hut as described above ensures that the spell does not contradict itself. The spell also says:
Creatures and objects within the dome when you cast this spell can move through it freely.
Prior to becoming animated, the caltrops are objects. Once animated, they become creatures. Either way, this part of the spell says that they can move through the dome freely. If creatures or objects bearing magical effects were barred from moving through the dome, then the spell would be contradicting itself. So it seems reasonable to assume that the intended interpretation is one that avoids this contradiction. Hence, the clause about blocking magic is most likely not intended to limit the mobility of creatures or objects through the dome, which the spell explicitly grants.
The caltrop swarm will not collapse the hut
The tiny hut spell also includes this limitation:
Nine creatures of Medium size or smaller can fit inside the dome with you. The spell fails if its area includes a larger creature or more than nine creatures.
This looks like it might be a problem, since after casting animate objects the hut contains at a minimum you and 10 animated caltrops, a total of 11 creatures (of course you can choose to animate fewer caltrops, but don't bother; keep reading). However, note that the spell uses the word "fails", not "ends". "Fail" is consistently used to describe spells that are cancelled prior to casting without ever taking effect, while "end" is used to describe a spell that is cancelled after it has already been cast and taken effect. (For example, compare the text of counterspell and dispel magic.)
So, like parents leaving for a weekend getaway and trusting their teenager not to throw a party while they're gone, the spell only checks the occupancy when it is first cast. After that, you can add as many creatures to the inside of the hut as you can fit, provided you have a way to get them inside.
Anyway, you probably weren't worrying about collapsing the hut until I brought it up, but you can go back to not worrying about it again.
You should check with your DM regardless
Regardless of the above, the question of exactly what is and is not blocked by the dome of a tiny hut spell is the subject of considerable debate. (Does it block sending? Scrying? Catapult?). The above is only my interpretation, and in the end, the only decision that matters is that of your DM.