Learn Hardcore Icelandic: Því miður man ég ekki lykilorðið. - Unfortunately I do not remember the password.

It’s a fixed adverbial meaning “unfortunately”. Literally it’s something like “for that, worse/less,” but you should treat it as a set phrase.
It most often appears at the start of a sentence, as in the example, but it can also be moved to the end: Ég man ekki lykilorðið, því miður.
Don’t change its form; use Því miður as-is (not “það miður,” etc.). At sentence start it’s capitalized: Því, otherwise því.

Why is the verb before the subject here (man ég), instead of ég man?

Icelandic main clauses are “V2” (verb-second). If you put something other than the subject first (here, Því miður), the finite verb (man) must come next, and then the subject (ég). Without that fronted phrase you’d say: Ég man ekki lykilorðið.

Where does the negation ekki go, and can it move?

In neutral word order, ekki comes after the finite verb: man ég ekki …
It normally appears before a full noun object: Ég man ekki lykilorðið.
With a pronominal object, the pronoun comes before ekki: Ég man það ekki (not: ✗ Ég man ekki það).
In questions, same idea: Manstu ekki lykilorðið?

Why is it lykilorðið with -ið at the end?

Icelandic attaches the definite article as a suffix. The base word is lykilorð (“password,” neuter). The definite singular is lykilorðið = “the password.” So the sentence says “I don’t remember the password,” i.e., a specific one already known in context.

What case is lykilorðið here, and why?

It’s the direct object of muna (“to remember”), which takes the accusative for “the thing remembered.” Since lykilorð is neuter, nominative and accusative look the same, so in the definite form you still see lykilorðið. For reference, the noun declines like this (singular): nom/acc lykilorð, dat lykilorði, gen lykilorðs (definite: lykilorðið, lykilorðinu, lykilorðsins).

How is muna (to remember) conjugated, and how is it different from munu (will)?

Present of muna: ég man, þú manst, hann/hún man, við munum, þið munið, þeir muna.
Simple past: ég mundi, þú mundir, hann mundi, við mundum, þið munduð, þeir mundu.
Supine/pp: munað (e.g., ég hef munað). Be careful not to confuse muna with munu (“will”). Some forms look identical in writing (e.g., við munum = “we remember” or “we will,” depending on context).

How do I ask “Do you remember the password?” naturally?

Use the contracted question form: Manstu lykilorðið? (from manst þú). Don’t write “mannstu.” Also note that Munt þú…? would mean “Will you…?”—that’s a different verb (munu).

Is “I forgot the password” expressed differently from “I don’t remember the password”?

Yes. “I forgot the password” is usually Ég gleymdi lykilorðinu. The verb gleyma (“to forget”) governs the dative, so the form is lykilorðinu (dat. def. sg.). “I don’t remember” is Ég man ekki (lykilorðið); “I have forgotten” can be Ég hef gleymt lykilorðinu or colloquially Ég er búinn/búin að gleyma lykilorðinu.

Can I use muna eftir instead of muna + accusative?

Both occur in real usage:

muna + accusative for remembering a thing is the simplest: Ég man lykilorðið.
muna eftir + dative is also common for recollecting someone/something: Ég man eftir því (“I remember it”), Ég man eftir henni (“I remember her”).
For “remember to (do/bring),” use muna að + infinitive or the imperative: Mundu að… / Mundu eftir + dat (e.g., Mundu eftir lyklinum! “Remember the key!”).

Is a comma needed after Því miður?

It’s optional. Many writers omit it: Því miður man ég ekki … You’ll also see Ég man ekki …, því miður with a comma when it’s sentence-final.

Any quick pronunciation tips for this sentence?

Þ (thorn) = unvoiced “th” as in “think”: Því ≈ “thvee.”
ð (eth) = voiced “th” as in “this”: miður ≈ “MITH-ur,” lykilorðið has a soft “th” sound near the end.
ekki: the “kk” is aspirated; think “EH-kyi” (the k is a bit stronger than in English).
ég is usually like “yeh(g),” with a soft final sound. Don’t worry about being perfect—aim for clear “th” sounds on Þ/ð and you’ll be understood.

Are there synonyms for lykilorð?

Yes. You may see aðgangsorð (“access word”) as a near-synonym. Lykilorð is by far the most common everyday term for “password.”

Can I move things around for emphasis?

Moving Því miður to the end is fine: Ég man ekki lykilorðið, því miður.
Fronting the object is possible for strong emphasis: Lykilorðið man ég ekki, but that’s marked and not the neutral choice.
Keeping the neutral order is safest: [topic] – verb – subject – adverbs (like ekki) – object: Því miður man ég ekki lykilorðið.

How do I say “I can’t remember the password” or “Now I remember it”?

“I can’t remember the password”: Ég get ekki munað lykilorðið or simply Ég man ekki lykilorðið.
“Now I remember it”: Nú man ég það. (Note pronoun before ekki/now if you negate or add adverbs: Nú man ég það ekki = “Now I don’t remember it.”)