Leikvöllurinn er bakvið kirkjuna, en kaffihúsið er rétt framhjá torginu.

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

So here:

• leikvöllurinn = the playground
• kaffihúsið = the café
• kirkjuna = the church
• torginu = the square

The exact ending changes depending on the noun’s gender, number, and case.

The base forms are:

• leikvöllur = playground
• kirkja = church
• kaffihús = café / coffeehouse
• torg = square

The forms in the sentence are changed because of definiteness and case.

Because after bakvið in this sentence, the noun is in the accusative.

• kirkja = base form
• kirkjuna = definite accusative singular
• kirkjunni = definite dative singular

So kirkjuna is the form that fits this prepositional phrase. In Icelandic, you usually have to learn prepositions together with the case they use.

Because framhjá takes the dative.

So:

• torg = base form
• torginu = definite dative singular

A very useful learning habit is to memorize prepositions with their case, for example framhjá + dative.

Here rétt means just, right, or immediately.

So rétt framhjá torginu means just past the square or right past the square.

It does not mean correct here. It is being used as an adverb to strengthen the location phrase.

In this sentence, yes. But you may also come across bak við and fram hjá in other materials.

For a learner, the main thing is to recognize the expressions and know what they mean and what case they require. Then it is best to follow the spelling convention used by your course, teacher, or dictionary.

Yes, in this sentence it is very similar to English.

Both clauses follow the pattern:

subject + er + place expression

• Leikvöllurinn er bakvið kirkjuna
• kaffihúsið er rétt framhjá torginu

The second er is necessary because en joins two full clauses.

You would use the indefinite forms, but the case patterns would stay the same:

Leikvöllur er bakvið kirkju, en kaffihús er rétt framhjá torgi.

Notice:

• kirkju is still shaped by the preposition bakvið
• torgi is still dative after framhjá

So removing the does not remove the case system.

Hjá usually means by, with, or at someone’s place, depending on context.

Framhjá means past, by, or just beyond, with more of a sense of passing something or being beyond it.

So in this sentence, the café is not merely by the square; it is just past the square.

Yes. Kaffihús is literally coffee house.

In normal English, though, the most natural translation is often café or coffee shop, depending on context. So kaffihúsið is literally the coffee house, but the café is usually the best everyday translation.