Hún spilar bæði á fiðlu og hljómborð.

It means both ... and ....

So:

• bæði á fiðlu og hljómborð = both the violin and keyboard

This is a very common Icelandic pattern:

• bæði X og Y = both X and Y

In Icelandic, the normal way to say that someone plays an instrument is:

• spila á + instrument

So:

• spila á fiðlu = play the violin
• spila á hljómborð = play keyboard / play a keyboard instrument

Literally, it looks a bit like play on an instrument, but that is just how Icelandic expresses this idea.

It can be written only once because it applies to both nouns.

So this sentence is understood as:

• Hún spilar bæði á fiðlu og á hljómborð.

Repeating the second á is possible, but leaving it out is natural and common when two items are joined together.

Spilar is the 3rd person singular present tense of spila (to play).

That means:

• ég spila = I play
• þú spilar = you play
• hún spilar = she plays

In context, Icelandic present tense can also sometimes match English is playing, depending on the situation.

With spila á in the sense of playing an instrument, the noun is normally in the accusative.

So here:

• fiðlu = accusative singular of fiðla
• hljómborð = accusative singular of hljómborð

A useful detail: fiðlu does not visibly show the accusative very clearly, because this noun has the same form in several singular oblique cases. And hljómborð looks the same in nominative and accusative. So you have to know the construction, not just rely on the form.

Icelandic often uses the bare noun when talking about playing an instrument in a general sense.

So:

• spila á fiðlu does not mean one specific violin
• it means playing violin as an instrument or skill

If you wanted to say the violin specifically, Icelandic would normally use the suffixed definite article:

• fiðlan = the violin
• hljómborðið = the keyboard

But that is not what is meant here.

Normally, no. In standard Icelandic, with musical instruments you usually need á after spila:

• spila á fiðlu
• spila á píanó
• spila á gítar
• spila á hljómborð

So leaving out á would sound wrong or at least nonstandard in this meaning.

Here it means a musical keyboard, such as an electronic keyboard.

That is different from a computer keyboard, which is usually:

• lyklaborð

So:

• hljómborð = musical keyboard
• lyklaborð = computer keyboard

No. You can also hear leika á with instruments:

• leika á fiðlu
• leika á píanó

But spila á is extremely common in everyday modern Icelandic, and it is the most useful pattern for learners to know first.