Teskeiðin datt af borðinu.

They are:

• teskeið — teaspoon (a feminine noun)
• detta — to fall
• af — off / from
• borð — table (a neuter noun)

So the sentence is built from very common Icelandic pieces: a noun, a past-tense verb, a preposition, and another noun.

Because Icelandic usually puts the definite article onto the end of the noun instead of using a separate word like English the.

• teskeið = teaspoon
• teskeiðin = the teaspoon

Here, the ending -in is the definite article attached to a feminine singular noun in the nominative.

Teskeiðin is nominative singular because it is the subject of the sentence — it is the thing doing the falling.

A good way to see that is to ask: What fell off the table?
Answer: The teaspoon.

That makes teskeiðin the subject, so nominative is the expected case.

Yes. Datt is the past tense singular of detta.

Some useful forms are:

• detta — to fall
• dettur — falls / is falling
• datt — fell
• duttu — fell (plural subject)
• dottið — fallen

So:

• Teskeiðin dettur af borðinu = The teaspoon falls / is falling off the table
• Teskeiðin datt af borðinu = The teaspoon fell off the table

Because the preposition af takes the dative case, and borðinu is the dative singular definite form of borð.

Here is the contrast:

• borðið = the table in nominative/accusative
• borðinu = the table in dative

Since af requires dative, you get:

• af borðinu = off the table

not af borðið.

Because af means movement off or from a surface, while á means on.

Compare:

• Teskeiðin var á borðinu = The teaspoon was on the table
• Teskeiðin datt af borðinu = The teaspoon fell off the table

So á borðinu describes location, while af borðinu describes movement away from that surface.

Not exactly.

• af often means off or from a surface
• frá means from in a more general sense

In this sentence, af is the natural choice because a teaspoon falls off a table — that is, away from a surface.

So af borðinu is more idiomatic than frá borðinu here.

Icelandic usually does not use a separate word for the, and it has no true indefinite article like English a/an.

So:

• teskeið = a teaspoon or just teaspoon, depending on context
• teskeiðin = the teaspoon
• borði = a table (here in dative)
• borðinu = the table (here in dative)

Compare these:

• Teskeiðin datt af borðinu = The teaspoon fell off the table
• Teskeið datt af borðinu = A teaspoon fell off the table
• Teskeiðin datt af borði = The teaspoon fell off a table
• Teskeið datt af borði = A teaspoon fell off a table

No, it does not have a direct object.

The structure is:

• Teskeiðin — subject
• datt — verb
• af borðinu — prepositional phrase

The verb detta is intransitive here, which means it does not take a direct object. The phrase af borðinu tells you where the falling happened from, but it is not an object.

Yes. Icelandic word order is flexible, but it follows the verb-second pattern in main clauses.

The neutral order here is:

• Teskeiðin datt af borðinu

But you can also front another part for emphasis or style:

• Af borðinu datt teskeiðin

That is still grammatical. The key point is that the finite verb datt stays in the second position.

A few helpful points:

• Icelandic stress is usually on the first syllable: TESkeiðin, BORðinu
• ei in skeið is roughly like the vowel in say
• ð is usually like the th in this
• tt in datt may sound sharper or a little breathy to English ears, often closer to daht than a plain English dat

A rough learner-friendly guide could be:

• Teskeiðin ≈ TES-skay-thin
• datt ≈ daht
• af borðinu ≈ av BOR-thi-nu

Those are only approximations, but they can help you get started.