Learn Hardcore Icelandic: Á skiltinu stendur að það sé bannað að keyra hratt hér. - On the sign it says that it is forbidden to drive fast here.

In Icelandic, the definite article is usually a suffix, not a separate word.
Because á here expresses location (where something is), it takes the dative case. So we get:

á + skilti (the sign, dative definite) → á skiltinu = on the sign.

If we were talking about movement onto the sign, we’d use the accusative:
á skiltið = onto the sign (accusative definite).

Why does the sentence start with Á skiltinu stendur instead of something like Það stendur á skiltinu?

Both word orders are possible:

Á skiltinu stendur að það sé bannað að keyra hratt hér.
Það stendur á skiltinu að það sé bannað að keyra hratt hér.

Starting with Á skiltinu just emphasizes where this information is found — on the sign. Icelandic word order is fairly flexible, and it’s common to put a location or time phrase first to set the scene.

So the difference is mostly about emphasis and style, not grammar:

Á skiltinu stendur… ≈ On the sign, it says that… (emphasis on the sign)
Það stendur á skiltinu… ≈ It says on the sign that… (more neutral order)

Why is stendur used here? Doesn’t it mean “stands”, not “says”?

Literally, standa means “to stand”, and stendur is “stands” (3rd person singular, present).

However, Icelandic often uses standa idiomatically when talking about written text:

Í bókinni stendur… = In the book it says…
Í samningnum stendur… = In the contract it says…
Á skiltinu stendur… = On the sign it says…

So in contexts involving written information, stendur is best translated as “it says” rather than literally “stands”.

Why is it að það sé bannað and not að það er bannað?

sé is the subjunctive form of vera (to be); er is the indicative form.

Icelandic often uses the subjunctive in clauses that report what something says, claims, or states (indirect speech / reported content).
Since Á skiltinu stendur… is basically “the sign says…”, the verb in the that-clause tends to be subjunctive:

Hann segir að hann sé þreyttur.
He says (that) he is tired. (subjunctive sé, because it’s what he says)

In your sentence:

Á skiltinu stendur að það sé bannað að keyra hratt hér.
= On the sign it says that it is forbidden to drive fast here.

Using er here is understandable but sounds less natural and less “correct” in standard Icelandic; sé is what learners are expected to use in this structure.

What is the það doing in að það sé bannað? Is it necessary?

Here það is a dummy/impersonal “it”, like English “it is forbidden”.

það = “it” (dummy subject)
sé bannað = “(is) forbidden” (neuter singular)

Icelandic likes to have a grammatical subject, even when there is no real, concrete subject. So we use það:

Það er kalt. = It is cold.
Það er bannað að reykja hér. = It is forbidden to smoke here.

You could occasionally hear að sé bannað að keyra hratt hér, but leaving out það sounds clipped or less natural. Standard, natural Icelandic is with það:

… að það sé bannað að keyra hratt hér.

What exactly is bannað here? Is it a verb form or an adjective?

bannað is the neuter singular past participle of the verb banna (to forbid).

In this sentence it behaves like a predicate adjective:

að eitthvað sé bannað = that something is forbidden

Why neuter singular? Because the “thing” that is forbidden is the action expressed by the infinitive clause að keyra hratt hér, and that’s treated as an abstract, neuter “it” (það).

Compare:

Það er bannað að reykja. = It is forbidden to smoke.
Það er bannað að fara inn. = It is forbidden to go in.

So structurally you have:

það (dummy subject, neuter)
sé (subjunctive of vera)
bannað (neuter predicate agreeing with that dummy “it”)

There are two að in the sentence. Are they the same “word” grammatically?

They look the same but function differently.

að það sé bannað…
Here að is a subordinating conjunction, like English “that”, introducing a clause:
- stendur að… = it says that…
…að keyra hratt hér
Here að is an infinitive marker, like “to” in English:
- að keyra = to drive

So:

First að = “that” (joins clauses)
Second að = “to” (forms the infinitive)

What does keyra mean exactly, and is it different from aka?

keyra means “to drive”, usually a car or vehicle.

There are two common verbs for “to drive”:

aka – older, more formal/literary; still used (e.g. in traffic laws, formal writing).
keyra – very common in everyday speech.

Examples:

Ég keyri í vinnuna. = I drive to work. (everyday)
Ekki aka of hratt. = Do not drive too fast. (sounds more formal)

In your sentence:

að keyra hratt hér = to drive fast here
Using aka instead would be possible but would sound more formal or legalistic:
að aka hratt hér.

Why is it hratt and not something like hraður?

hraður is an adjective: fast, quick (describing a noun).
hratt is its adverb form: fast, quickly (describing a verb).

You need an adverb to describe how you drive:

að keyra hratt = to drive fast / to drive quickly.

Compare:

hraður bíll = a fast car (adjective, describing bíll)
Bíllinn keyrir hratt. = The car drives fast. (adverb, describing keyrir)

Why is hér used, and how is it different from hérna or þarna?

hér means “here” in a fairly neutral or slightly formal way.
In this sentence, it refers to this place / this area where the sign applies.

Differences:

hér – here (neutral, often used in writing, signs, more formal speech)
hérna – right here / around here (more colloquial, often spoken)
þarna – there (a place near but not where the speaker currently is)

A sign will almost always use hér, not hérna, because hér sounds more appropriate and formal for written notices:

Bannað að reykja hér. = Smoking is forbidden here.
Hvað á að gera hérna? = What are we supposed to do here (around here)? (more conversational)

So hratt hér is perfectly natural for a sign that is talking about this location.