Mér var sýnt hvar sviðið var áður en fólkið kom inn í salinn.

Because sýna normally takes the person receiving the information in the dative:
að sýna einhverjum eitthvað = to show someone something.

So in this sentence, mér means to me or me in the dative. Icelandic keeps that dative form here, even though English says I was shown.

A useful literal breakdown is:

• Mér = to me
• var sýnt = was shown

So Mér var sýnt ... is literally something like To me was shown ...

It is a passive construction, and more specifically an impersonal passive-style pattern.

The active version would be:

• Einhver sýndi mér hvar sviðið var ...
• Someone showed me where the stage was ...

In the sentence you have, the person doing the showing is omitted, and the focus is on the experience of me:

• Mér var sýnt ... = I was shown ...

This is very common in Icelandic when the agent is unknown, unimportant, or just not mentioned.

Because the participle here appears in the neuter singular, which is the normal default form in this kind of construction when there is no clear nominative subject for it to agree with.

The thing being shown is not a simple noun here, but the clause:

• hvar sviðið var = where the stage was

Since the sentence is built impersonally, Icelandic uses the default neuter singular participle:

• var sýnt

So sýnt is not referring to the gender of mér. It is just the standard form used in this structure.

It is an embedded question or indirect question, and it functions as the content of what was shown.

So the sentence does not mean that the stage itself was physically shown in focus as a noun object. Rather, what was shown was the information:

• where the stage was

You can think of it as:

• I was shown [where the stage was].

So the whole clause hvar sviðið var acts like the object of sýnt.

Because hvar sviðið var is a subordinate clause or indirect question, not a direct question.

In a direct question, Icelandic often has verb-first order:

• Hvar var sviðið?
• Where was the stage?

But in an indirect question, the word order is more like a statement:

• hvar sviðið var
• where the stage was

So this is the normal subordinate-clause order.

They do two different jobs:

• Mér var sýnt: here var is the auxiliary was used to build the passive.
• hvar sviðið var: here var is simply the past tense of vera = to be, meaning was.

So the first var helps form was shown, while the second var means was in where the stage was.

Áður en means before when followed by a full clause.

Here:

• áður en fólkið kom inn í salinn
• before the people came into the hall

It introduces a time clause. After áður en, you get a normal finite verb clause:

• fólkið kom inn í salinn

So this structure is very similar to English before + clause.

Because fólk in Icelandic is a collective noun. Grammatically, it is singular neuter, even though in English it often corresponds to people.

So Icelandic says:

• fólkið kom
• literally the people came, but grammatically the people is treated as singular

That is why the verb is singular:

• kom = singular past

This is normal with fólk.

Because Icelandic uses different case patterns to distinguish movement from location.

Here, the people are moving into the hall:

• kom inn í salinn
• came into the hall

That is why salinn is in the accusative.

Compare:

• í salnum = in the hall / inside the hall
  This would describe location, not motion.

So:

• inn í salinn = motion into the hall
• í salnum = being in the hall

These are all nouns with the definite article attached to the end, which is very common in Icelandic.

• svið = stage

• sviðið = the stage

• fólk = people

• fólkið = the people

• salur = hall
• salinn = the hall
  Here the form is accusative singular because of movement: inn í salinn

So instead of a separate word like English the, Icelandic often adds the article as an ending.

Yes. A natural active version would be:

• Einhver sýndi mér hvar sviðið var áður en fólkið kom inn í salinn.
• Someone showed me where the stage was before the people came into the hall.

The passive version removes the person doing the showing, which can sound more neutral or more natural if the agent does not matter.

Because hvar sviðið var is an indirect question, not a plain that-clause.

Icelandic uses question words like:

• hvar = where
• hvenær = when
• hvernig = how
• af hverju = why

without að in this kind of construction.

So after sýna, you can say:

• sýna mér hvar ...
• show me where ...

not:

• sýna mér að hvar ...

That would be wrong.

Yes, in meaning it is very close.

• koma inn í = come into, go in, enter

So:

• fólkið kom inn í salinn
• the people came into the hall
• the people entered the hall

In English, enter is often more formal, while Icelandic commonly uses this kind of verb + direction expression.