Á leiðinni heim spurði hún hvort við gætum farið aftur í íþróttahúsið næsta laugardag.

Because Icelandic normally follows a verb-second pattern in main clauses.

Here, Á leiðinni heim is placed first. Once that adverbial phrase takes first position, the finite verb spurði must come second, so the subject hún comes after it:

Á leiðinni heim spurði hún ...

If you started the sentence with the subject instead, then you would get:

Hún spurði ...

So this is normal Icelandic word order, not an inversion for emphasis in the English sense.

Literally, it is something like on the way home.

• á = on
• leiðinni = the way, in the dative singular
• heim = homeward, toward home

The noun is leið. After á in this kind of expression, Icelandic uses the dative, so leið becomes leiðinni.

This phrase is very common and idiomatic:

• á leiðinni = on the way
• á leiðinni heim = on the way home

The -inni ending shows both dative and definite article.

Because heim and heima are different:

• heim = to/toward home (direction)
• heima = at home (location)

In á leiðinni heim, the idea is movement in the direction of home, so heim is the correct form.

Compare:

• Ég fer heim. = I go home.
• Ég er heima. = I am at home.

That same directional idea is why heim appears here.

Here hvort introduces an indirect yes/no question, so it means whether or if.

After a verb like spyrja meaning to ask, Icelandic very often uses hvort:

• Hún spurði hvort við gætum farið ... = She asked whether / if we could go ...

This is different from a direct question:

• Getum við farið ... ? = Can we go ... ?

So hvort is the marker that turns the clause into an embedded yes/no question.

In standard Icelandic, hvort is the usual word for whether / if in an indirect question.

So after spurði meaning asked, hvort is the most natural choice:

• spurði hvort ... = asked whether ...

By contrast, ef is more commonly used for if in the sense of a condition:

• Ef við höfum tíma, förum við. = If we have time, we go.

So a good learner rule is:

• hvort = whether / if in indirect questions
• ef = if in conditions

Gætum is the past subjunctive form of geta for við.

That form is very common in reported speech or indirect questions, especially after a past-tense verb like spurði.

Compare:

• Direct question: Getum við farið?
• Reported version: Hún spurði hvort við gætum farið.

So the sentence is using the normal kind of backshift you often see after she asked in English, where English might also use could instead of can.

It also sounds a bit less direct or more tentative, which fits well after spurði hvort.

After geta in this meaning, Icelandic normally uses the form farið.

So the standard pattern is:

• get farið
• gat farið
• gætum farið

This form is traditionally called sagnbót. For many learners, the important thing is simply to remember the pattern:

• Ég get farið.
• Við gætum farið.

Even though dictionaries usually list the verb as fara, after geta you normally get farið.

Because Icelandic uses:

• accusative after í when there is motion into
• dative after í when there is location in

Here the idea is go into / go to the sports hall, so the accusative is used:

• í íþróttahúsið

If the meaning were be in the sports hall, it would be:

• í íþróttahúsinu

So:

• fara í húsið = go into the building
• vera í húsinu = be in the building

That accusative-vs-dative contrast is very important in Icelandic.

Yes. Icelandic forms compounds very freely, and they are usually written as one word.

íþróttahúsið breaks down like this:

• íþrótta- = sports
• hús = house/building/hall
• -ið = the definite article

So it is literally something like the sports-building / the sports hall.

The first part íþrótta- comes from the plural/genitive form used in compounds. This is very common in Icelandic compound nouns.

Because Icelandic often uses the accusative of time for expressions like next Saturday, last week, all day, and so on.

So næsta laugardag means next Saturday without needing a preposition.

This is a common time-expression pattern:

• næsta laugardag = next Saturday
• síðasta vetur = last winter
• allan daginn = all day

You may also hear á laugardag for on Saturday, but with next Saturday, the bare accusative expression næsta laugardag is very natural.

It can suggest both, and that is normal.

aftur often means:

• again
• back

In this sentence, because they are talking about going to the sports hall another time, and probably returning to a place they have been before, English could express it as either:

• go again
• go back

So aftur here has that overlapping sense of another time / back to the same place.