Learn Hardcore Icelandic: Geturðu útskýrt þetta nánar?

Yes. "Geturðu" is the cliticized colloquial form of "getur þú" (verb + you-sg). Both are correct; "Geturðu" is what you’ll hear most in speech and see often in writing. Icelandic frequently cliticizes 2nd person singular þú after a verb:

ert þú → ertu
vilt þú → viltu
getur þú → geturðu This is normal and not informal in a negative way—just natural.

How do I pronounce the tricky letters here (ð, þ, ú, ý, á), and the whole sentence?

ð (in "Geturðu") sounds like English "th" in this.
þ (in "þetta") sounds like "th" in thing.
ú is like long "oo" in goose.
ý sounds like long "ee" (lengthened i-sound).
á is like the diphthong "ow" in now. A rough, friendly approximation: GET-ur-thu OOT-skeer-t THET-ta NOW-nar. Stress the first syllable of each word.

Why is it "útskýrt" and not the infinitive "útskýra"?

With the verb geta (can/be able), Icelandic uses the verb’s supine (which looks like the neuter past participle):

Ég get komið (I can come)
Þú getur séð (you can see)
Geturðu útskýrt (can you explain) Other modals, like vilja (want) or mega (may), take the infinitive:
Ég vil útskýra þetta.
Má ég útskýra þetta?

What exactly is "þetta," and why that form?

Þetta is the neuter singular demonstrative "this," here in the accusative as the direct object of "útskýra." If the object were masculine or feminine, you’d see forms like þennan (masc acc) or þessa (fem acc). Neuter often appears with abstract "this" referring to a situation or idea.

What does "nánar" add? How is it different from "meira," "betur," or "frekar"?

nánar = "in more detail," "more closely" (depth/precision).
meira = "more" (quantity/amount).
betur = "better" (quality/improvement).
frekar = "rather/instead" (preference or alternative). So "útskýra þetta nánar" asks for a more detailed explanation, not necessarily a longer one.

Is "Geturðu..." polite enough? How do I say "Could you…?" or add "please"?

"Geturðu..." is neutral and fine. To soften it:

Gætirðu útskýrt þetta nánar? (conditional "could you")—more polite.
Add softeners like kannski (maybe) or aðeins (a bit): "Gætirðu kannski útskýrt þetta aðeins nánar?" "Vinsamlegast" ("please") is common in signs/emails but can sound formal or officious in casual speech. Softeners and the conditional are more natural in conversation.

How do I negate this? Where does "ekki" go?

Put ekki right after the finite verb (plus the clitic if present):

Geturðu ekki útskýrt þetta nánar? (Can’t you explain this in more detail?) Without clitic: Getur þú ekki... also works.

Can I move "nánar" around?

The most natural spot is at the end: útskýrt þetta nánar. You can also say útskýra nánar if there’s no object. Fronting "nánar" (e.g., "nánar útskýrt þetta") is uncommon in ordinary speech and sounds marked or stylistic. Keep it at the end for everyday usage.

How do I say this to more than one person?

Use 2nd person plural:

Standard: Getið þið útskýrt þetta nánar?
Common colloquial contraction: Getiði útskýrt þetta nánar? Both are understood; "getið þið" is the safe, standard choice.

How do I add "to me/for us," as in "explain it to me"?

"Útskýra e-ð fyrir e-m" is the typical pattern:

Geturðu útskýrt þetta nánar fyrir mér? (for me)
... fyrir okkur/þeim/honum/henni etc. You’ll also hear double-object patterns like "útskýra mér þetta," but "fyrir + dat." is the safest.

What are the stress and rhythm rules I should know here?

Icelandic stresses the first syllable of content words. So: GEturðu ÚTskýrt ÞETta NÁNar. Keep vowels long in open syllables and before certain consonant environments, but as a learner, first nail the initial-stress pattern and clear consonants.

Any spelling gotchas?

y/ý and i/í often sound the same; here it’s útskýrt (with ý), not "útskírt."
Remember þ vs ð: "þetta" with þ, "Geturðu" ends with -ðu.
Writing the clitic together is standard: Geturðu (though Getur þú is also fine).

What is "útskýra" built from?

It’s út- + skýra. "Skýr" = clear; "skýra" = to make clear; "útskýra" = to explain (literally "out-clarify"). The prefix is not separable in Icelandic (unlike separable prefixes in German).

How is a yes/no question formed here?

By placing the finite verb first. Statement order would be "... þú getur útskýrt ...", but the question is Geturðu útskýrt ...? No extra do-support or question particle is needed—just verb-first word order.