Hornhlífin er á hurðarkarminum við beygluna.

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

• hornhlíf = corner protector / corner guard
• hornhlífin = the corner protector

In this sentence, hornhlífin is the subject, so it is in the nominative singular definite form.

This is because hurðarkarmur is a compound noun:

• hurð = door
• karmur = frame

In many Icelandic compounds, the first noun appears in a form that comes from the genitive. So:

• hurð → hurðar-
• hurðarkarmur = door frame literally frame of a door

This is very common in Icelandic, and learners often just need to memorize the compound as a whole.

Because it follows á in a location meaning.

With á, Icelandic often works like this:

• á + dative = location, on / in / at
• á + accusative = movement onto something

Here the sentence says where something is, not where it is moving:

• Hornhlífin er á hurðarkarminum = The corner protector is on the door frame

So the noun is dative:

• hurðarkarmi = dative singular
• hurðarkarminum = dative singular definite, on the door frame

A useful comparison:

• Hún er á hurðarkarminum. = It is on the door frame.
• Hún fer á hurðarkarminn. = It goes onto the door frame.

Because við normally takes the accusative in modern Icelandic.

So:

• beygla = dent
• beygluna = the dent in the accusative singular definite

That is why the phrase is:

• við beygluna = by the dent / next to the dent

This is something learners often have to memorize with prepositions: different prepositions require different cases.

It means something like:

• by the dent
• next to the dent
• beside the dent

So it tells you the corner protector’s position relative to the dent.

It does not necessarily mean direct contact in the way English against sometimes does. It is mainly a nearby-position phrase.

No—this is a very normal Icelandic main-clause word order.

• Hornhlífin = subject
• er = verb
• á hurðarkarminum við beygluna = location phrase

So the structure is basically:

subject + verb + place

That said, Icelandic is fairly flexible because the finite verb usually stays in second position. For example, you could also say:

• Á hurðarkarminum við beygluna er hornhlífin.

That sounds more like emphasizing the location.

If you made the nouns indefinite, you would remove the suffixed definite article endings:

• Hornhlíf er á hurðarkarmi við beyglu.

That would mean something more like:

• A corner protector is on a door frame by a dent.

In practice, the original definite version sounds much more natural if you are talking about specific, known things.

Yes, very normal. Icelandic uses compound words all the time.

For example:

• horn + hlíf → hornhlíf
• hurðar + karmur → hurðarkarmur

Then grammar gets added to the end:

• hornhlíf → hornhlífin
• hurðarkarmur → hurðarkarminum

So a long Icelandic word is often just:

compound noun + case/definiteness ending

That can look intimidating at first, but it becomes much easier once you start spotting the parts inside the word.