Skærin eru á borðinu, en límbandið er í skúffunni við tölvuna.

Because the verb vera (to be) agrees with the subject in number.

• skærin = the scissors → grammatically plural
• límbandið = the tape → grammatically singular

So:

• skærin eru = the scissors are
• límbandið er = the tape is

This is very similar to English: scissors are, but tape is.

Skæri is normally a plural-only noun in Icelandic, just like scissors in English is normally treated as plural.

So:

• skæri = scissors
• skærin = the scissors

That is why the sentence uses the plural verb eru.

If you are thinking of one pair of scissors, Icelandic still normally uses the plural noun.

These endings show things like the definite article (the) and also case.

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

Here are the forms in the sentence:

• skærin = the scissors
• límbandið = the tape
• borðinu = the table (in the form required after á here)
• skúffunni = the drawer (in the form required after í here)
• tölvuna = the computer (in the form required after við)

So these endings are not random: they are part of Icelandic grammar, showing definiteness, case, and sometimes number.

Because Icelandic usually puts the onto the end of the noun as a suffix.

Compare:

• borð = table

• borðið = the table

• skúffa = drawer
• skúffan = the drawer

In this sentence, the nouns are also changing for case, so you get forms like:

• borðinu
• skúffunni
• tölvuna

English keeps the separate, but Icelandic usually does not.

Because á and í often take the dative when they describe location rather than movement.

Here the sentence is saying where something is:

• á borðinu = on the table
• í skúffunni = in the drawer

That is a static location, so Icelandic uses dative after these prepositions.

A very useful contrast is:

• á borðinu = on the table (location)
• á borðið = onto the table (movement)

and

• í skúffunni = in the drawer (location)
• í skúffuna = into the drawer (movement)

Because við is different from á and í here.

The preposition við normally takes the accusative, including in meanings like by, next to, or beside.

So:

• við tölvuna = by the computer / next to the computer

Even though it expresses location in English, Icelandic still uses the case required by við, which is accusative.

This is something learners usually just have to memorize with the preposition.

Here is the case breakdown:

• skærin — nominative plural
  It is the subject of the first clause.

• borðinu — dative singular
  Used after á for static location.

• límbandið — nominative singular
  It is the subject of the second clause.

• skúffunni — dative singular
  Used after í for static location.

• tölvuna — accusative singular
  Used after við.

This kind of case-mapping is very common in Icelandic sentences with prepositions.

Yes, they matter because gender affects endings.

The nouns here are:

• skæri — neuter (plural-only in normal use)
• borð — neuter
• límband — neuter
• skúffa — feminine
• tölva — feminine

That is part of why the endings differ:

• neuter: límbandið, borðinu
• feminine: skúffunni, tölvuna

So when learning Icelandic nouns, it is very helpful to learn each noun together with its gender.

Yes. In this sentence, en simply means but and joins two main clauses:

• Skærin eru á borðinu
• en límbandið er í skúffunni við tölvuna

So the whole sentence means one thing is in one place, but the other thing is somewhere else.

This is a very common and straightforward use of en.

Because Icelandic makes heavy use of compound nouns.

Límband is made from:

• lím = glue
• band = strip / band

Together, límband means adhesive tape / tape.

Then the definite article is added to the whole compound:

• límband = tape
• límbandið = the tape

This is very typical of Icelandic. English often writes similar ideas as one word, two words, or with a hyphen, but Icelandic strongly prefers compounds.

Then the nouns after á and í would usually switch to the accusative.

Current sentence:

• Skærin eru á borðinu = the scissors are on the table
• límbandið er í skúffunni = the tape is in the drawer

If you were talking about putting them there, you would get:

• Settu skærin á borðið = Put the scissors onto the table
• Settu límbandið í skúffuna = Put the tape into the drawer

So a very important Icelandic pattern is:

• dative for location
• accusative for movement toward/into/onto

with prepositions like á and í.