Learn Hardcore Icelandic: Bankaðu ekki of fast; barnið sefur. - Don’t knock too hard; the child is sleeping.

-ðu is an enclitic form of the 2nd-person singular pronoun þú (you) attached to the imperative. So banka is the imperative stem (Knock!) and bankaðu is essentially Knock, you! In modern Icelandic, this attachment is very common in positive imperatives and often sounds more natural/complete than the bare imperative on its own.

Is bankaðu the only way to say Knock!?

No. You can also say:

Banka! (bare imperative; can sound more abrupt)
Bankaðu! (very common, neutral) For plural or formal address:
Bankið! (to more than one person / polite plural)

Why is the negation ekki placed after bankaðu?

In Icelandic, the negator ekki typically follows the finite verb in main clauses, including imperatives. So the normal pattern is:

Verb + ekki + ... Here: Bankaðu ekki ...

How would I form the negative imperative without -ðu?

You can, but it tends to sound more clipped:

Banka ekki of fast. The more common everyday phrasing is still:
Bankaðu ekki of fast. For plural:
Bankið ekki of fast.

What does of do here, and why not just use mjög?

of means too/excessively, implying it’s more than appropriate. mjög means very and does not necessarily imply excess. So:

of fast = too hard / excessively hard
mjög fast = very hard (could still be intended)

Is fast an adjective or an adverb here? Does it change form?

Here fast functions as an adverb meaning hard/firmly (how you knock). In this usage it doesn’t inflect for gender/number/case. Many Icelandic adverbs look identical to neuter adjective forms, but in this sentence it’s best understood as an adverb.

Does banka need a preposition, like banka á?

Often, yes—especially if you mention what you’re knocking on:

Banka á dyrnar. = knock on the door
But it’s also normal to omit the object when it’s obvious from context:
Bankaðu ekki of fast. (understood: on the door)

Why is it barnið and not barn?

barnið is barn (child) with the suffixed definite article -ið, so it means the child. Icelandic usually marks definiteness on the noun itself:

barn = a child
barnið = the child

What case is barnið in here?

It’s nominative, because barnið is the subject of the verb sefur:

Barnið sefur. = the child sleeps/is sleeping

What tense is sefur, and does it mean sleeps or is sleeping?

sefur is present tense of sofa (to sleep). Icelandic present tense often covers both:

sleeps (habitual) and
is sleeping (right now)
Context decides; with a warning like this, it’s typically understood as is sleeping.

Why is there a semicolon (;) instead of og or a comma?

The semicolon links two closely related independent clauses: a request/warning and its reason. You could also write:

Bankaðu ekki of fast, barnið sefur. (comma possible)
Bankaðu ekki of fast því að barnið sefur. (because)
Bankaðu ekki of fast. Barnið sefur. (two sentences)

How would this change if I were speaking to more than one person?

Use the plural imperative:

Bankið ekki of fast; barnið sefur.
Here bankið is the 2nd-person plural imperative (also used as a polite form in some contexts).

How do you pronounce the tricky parts: bankaðu, ekki, barnið?

A practical guide:

bankaðu: stress on the first syllable BAN-, with -aðu sounding like a soft ending (the ð is very light, often like a voiced th in this, and may be barely heard in fast speech).
ekki: the kk is a strong unvoiced sound; it’s not like English egg.
barnið: again stress on the first syllable BAR-; the ð is typically very soft.