Staðgreiðslan er á launamiðanum.

It is:

subject + verb + prepositional phrase

• Staðgreiðslan = the subject
• er = is
• á launamiðanum = on the payslip

So the grammar is very close to English: The withholding tax is on the payslip.

Because Icelandic usually adds the definite article to the end of the noun instead of using a separate word like the.

• staðgreiðsla = withholding tax / tax withheld at source
• staðgreiðslan = the withholding tax

This noun is feminine, and in the nominative singular the definite form is staðgreiðslan.

It is nominative singular because it is the subject of the sentence.

In Icelandic, the subject of a simple sentence with vera (to be) is normally in the nominative:

• Staðgreiðslan er ... = The withholding tax is ...

Er is the 3rd person singular present tense of vera (to be).

Since staðgreiðslan is singular, Icelandic uses singular er:

• ég er = I am
• þú ert = you are
• hann/hún/það er = he/she/it is

So Staðgreiðslan er ... means The withholding tax is ...

The base form is launamiði, which means payslip or pay slip.

Here it appears as launamiðanum, which is the dative singular definite form:

• launamiði = a payslip
• launamiðanum = on the payslip

Because á can take different cases depending on the meaning.

Here it means location: something is on the payslip. With this meaning, á takes the dative.

So:

• á launamiðanum = on the payslip (location)

If there were movement onto something, Icelandic would often use the accusative instead:

• setja eitthvað á launamiðann = put something onto the payslip

So in this sentence, the tax is already there, so the dative is used.

No. Á is one of the Icelandic prepositions that can take more than one case.

A very useful rule is:

• dative for location: where something is
• accusative for movement toward a place: where something goes

In this sentence:

• er á launamiðanum = is on the payslip
  so it is location, and therefore dative.

Because Icelandic usually writes compound nouns as one word, much more often than English does.

For example:

• laun = wages, pay
• miði = slip, note, ticket
• launamiði = payslip

And:

• staðgreiðsla is also a compound noun, used for withholding at source or withholding tax

So if you are an English speaker, one common adjustment is getting used to Icelandic compounds being written together.

Because the sentence is talking about a specific item on a specific payslip, not about these things in general.

• staðgreiðslan = the withholding tax
• launamiðanum = the payslip

If you removed the definite article, the sentence would sound more general or less natural in this context.

Yes. Icelandic word order is flexible, but main clauses usually follow the verb-second rule.

So you can also say:

Á launamiðanum er staðgreiðslan.

That version puts more emphasis on where it is: On the payslip is the withholding tax.

Both are grammatical, but Staðgreiðslan er á launamiðanum is the more neutral order.