Ég set hvítan borðdúk á borðið.

Because borðdúk(ur) is a masculine noun, and here it’s the direct object of the verb set (to put/set), so it’s in the accusative singular:

• nominative: hvítur borðdúkur
• accusative: hvítan borðdúk
  The adjective has to match the noun in gender, number, and case.

Borðdúkur is the nominative singular (dictionary form). In the sentence it’s the object, so it takes accusative singular, which for many masculine nouns drops -ur:

• borðdúkur (nom.) → borðdúk (acc.)

With á, Icelandic uses:

• accusative for movement/direction onto something (putting it onto the table): á borðið
• dative for location/being on something (already on the table): á borðinu

So Ég set … á borðið focuses on the action of placing it.

Borðið means the table. Icelandic usually expresses the definite article as a suffix:

• borð = a table
• borðið = the table (-ið is the neuter singular definite ending)

They refer to different things:

• hvítan borðdúk = the object being moved (the tablecloth)
• á borðið = the destination (onto the table)

Icelandic often makes these roles clear through case (accusative on the object; accusative after á for direction).

Yes, that can be possible, but it changes focus. The neutral order is:

• Ég set hvítan borðdúk á borðið.

Putting the prepositional phrase earlier can sound more marked/contrastive, like emphasizing where you’re putting it.

• setja = the infinitive (to put/set)
• set = 1st person singular present tense (I put / I am putting)

So Ég set literally means I put.

It can mean either. Icelandic present tense often covers both:

• habitual/general: I put
• ongoing (depending on context): I am putting

Context usually makes it clear.

A learner-friendly approximation:

• Ég ≈ yegh (with a palatal y-like start)
• hvítan: the hv is often like kv (so roughly kvee-tan)
• borð has ð, which is like the th in this (but often very soft)
• á is a long ow/au-like vowel (as in loud, but longer)

If you want, tell me your dialect (US/UK/etc.) and I can give a tighter approximation.

Icelandic doesn’t have an independent word like English the in front of nouns. Definiteness is usually:

• indefinite: no article (so borðdúkur can mean a tablecloth)
• definite: a suffix (borðdúkurinn = the tablecloth)

In your sentence, borðdúk is indefinite: a white tablecloth.

You’d make the tablecloth definite too:

• Ég set hvíta borðdúkinn á borðið.

Notice the changes:

• borðdúkurinn (the tablecloth) becomes accusative borðdúkinn
• the adjective becomes weak (definite context): hvíta (not hvítan)