Ég set tuskuna í vaskinn eftir að ég þurrka borðið.

Setja is the infinitive, meaning to put. In the sentence, you need the finite present-tense form, because the subject ég is doing the action now or habitually.

So:

• að setja = to put
• ég set = I put
• þú setur = you put
• hann/hún/það setur = he/she/it puts

So Ég set ... is correct, not Ég setja ....

In Icelandic, the is usually added to the end of the noun instead of being a separate word.

In this sentence:

• tuska = cloth

• tuskuna = the cloth

• vaskur = sink

• vaskinn = the sink

• borð = table
• borðið = the table

So the endings are doing the job that the does in English.

Because Icelandic nouns change form for:

• gender (masculine, feminine, neuter)
• number (singular/plural)
• case (nominative, accusative, dative, genitive)
• and whether they are definite or indefinite

Here:

• tuskuna is feminine singular definite accusative
• vaskinn is masculine singular definite accusative
• borðið is neuter singular definite accusative

So the different endings are not random; they reflect each noun’s grammar.

They are in the accusative because they are the direct objects of the verbs.

• setja eitthvað = to put something
  → tuskuna is the thing being put

• þurrka eitthvað = to wipe something
  → borðið is the thing being wiped

A very common pattern in Icelandic is that the direct object of a verb is in the accusative.

Because Icelandic often uses different cases after prepositions depending on whether there is movement or location.

With í:

• í + accusative = into, motion toward the inside
• í + dative = in, static location

So:

• Ég set tuskuna í vaskinn = I put the cloth into the sink
• Tuskan er í vaskinum = The cloth is in the sink

That is one of the most important preposition patterns in Icelandic.

Because eftir að is used before a full clause with its own verb.

Here the clause is:

• ég þurrka borðið

So you need:

• eftir að ég þurrka borðið = after I wipe the table

By contrast, eftir by itself is used before a noun phrase:

• eftir kvöldmat = after dinner
• eftir vinnu = after work

So:

• eftir að + clause
• eftir + noun phrase

Yes, if you are using a normal finite clause, you need the subject again.

So:

• Ég set tuskuna í vaskinn eftir að ég þurrka borðið.

The second ég belongs to the clause ég þurrka borðið.

English often allows a shorter structure like after wiping the table, but Icelandic usually uses a full clause here. If you want a more compressed version, you can use something like:

• eftir að hafa þurrkað borðið = after having wiped the table

But eftir að ég þurrka borðið is very straightforward and natural.

Because the whole sentence is in the present and describes a habit, routine, or a general sequence.

So the idea is something like:

• I put the cloth in the sink after I wipe the table

This is not necessarily talking about one completed past event. It can describe what you normally do.

If you were talking about a past event, you would usually switch both verbs to the past:

• Ég setti tuskuna í vaskinn eftir að ég þurrkaði borðið.

So the present tense here is normal if the sentence describes a regular action or a future-like sequence.

Yes. You can say:

• Eftir að ég þurrka borðið set ég tuskuna í vaskinn.

That is also correct.

But notice what happens in the main clause: Icelandic keeps the finite verb in second position, so after the opening subordinate clause you get:

• set ég not
• ég set

That word order often surprises English speakers, but it is a basic feature of Icelandic main clauses.

Because borð is a neuter noun, and in Icelandic many neuter nouns have the same form in the nominative and accusative singular.

So:

• borðið can be nominative singular definite
• borðið can also be accusative singular definite

In this sentence, it is accusative because it is the object of þurrka.

This is very common with neuter nouns, so it is something you will see a lot.