Bankaðu ekki of fast; barnið sefur.

-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.

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)

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 ...

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.

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)

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.

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)

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

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

• Barnið sefur. = the child sleeps/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.

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)

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).

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.