Við kvörtum ekki, þó að röðin sé löng.

Kvörtum is the present tense, 1st person plural form of the verb kvarta (to complain).

• infinitive: að kvarta
• present (we): við kvörtum
  It’s a regular pattern for many Icelandic verbs: -um often marks we in the present tense.

In a normal main clause Icelandic has verb-second (V2) word order, so the finite verb comes early, and ekki typically comes right after the finite verb (or after the subject + finite verb).
So you get:

• Við kvörtum ekki (We do not complain)

You’ll also see ekki after an auxiliary if there is one, e.g. Við getum ekki kvarta (We cannot complain).

Þó að introduces a concessive subordinate clause, roughly though / even though. It sets up a contrast: the main clause holds true despite what follows in the þó að clause.

Also note: the comma before þó að is normal because it introduces a subordinate clause.

Sé is the present subjunctive of vera (to be). After þó að, Icelandic often uses the subjunctive, especially when the clause is concessive, slightly hypothetical, or presented as a circumstance being conceded.
So:

• indicative: er (is)
• subjunctive: sé

In some contexts you may hear the indicative after þó að, but þó að ... sé is a very common, standard choice.

It’s the finite verb in the subordinate clause and it agrees with the subject röðin (singular). Paradigm-wise, it’s from vera:

• indicative present: ég er, þú ert, hann/hún/það er, við erum, þið eruð, þeir/þær/þau eru
• subjunctive present (common set): ég sé, þú sért, hann/hún/það sé, við séum, þið séuð, þeir/þær/þau séu

Here you have: (þó að) röðin sé ...

Röðin means the line/queue. Icelandic often attaches the definite article to the end of the noun:

• röð = a line / a row / a queue
• röðin = the line

So -in here is the suffixed definite article for a feminine noun in the nominative singular.

Löng is feminine nominative singular to agree with röðin (a feminine noun) in a predicate adjective position:

• röðin (fem. nom. sg.) + löng (fem. nom. sg.)

Comparison with other genders:

• masculine: langur
• feminine: löng
• neuter: langt

Also, langa is the weak feminine form used mainly attributively with a definite noun (e.g. langa röðin = the long line). But after vera (predicate position), Icelandic typically uses the strong form: röðin er löng.

Yes: once you’re inside a subordinate clause (introduced by að here), Icelandic does not follow main-clause V2 in the same way. The finite verb commonly comes after the subject:

• þó að röðin sé löng (though the line is long)

That subject–verb order is typical for subordinate clauses.

Common points:

• þ in þó is like English th in thin (voiceless).
• ó is a long vowel, roughly like a long o (not exactly English oh, but close enough to start).
• ð in röðin is like English th in this (voiced), often softer between vowels.
• ö in röðin is similar to a rounded vowel like German ö (not found in standard English).
• ng in löng is typically a single ng sound like in English song.

Stress in Icelandic is almost always on the first syllable: RÖ-ðin, LÖng.

Yes, and the nuance can change slightly:

• Við kvörtum ekki is a straightforward statement: we do not complain (as a general fact / in this situation).
• Við erum ekki að kvarta is more like We’re not (in the act of) complaining, often used to emphasize that you’re not trying to complain right now, even if you’re describing a negative situation.

Both are natural; the original sentence is concise and very common.