Ég kveiki á lampanum þegar það er myrkur úti.

Kveiki is the 1st person singular present tense form of the verb kveikja (to light/turn on).

• Infinitive: að kveikja
• Present: ég kveiki, þú kveikir, hann/hún/það kveikir, við kveikjum, þið kveikið, þeir/þær/þau kveikja
  So Ég kveiki is simply the normal conjugation for I turn on / I light.

In this expression, kveikja á + dative is an idiomatic verb + preposition combination meaning to turn on (a light/device).
So kveikja by itself is more like ignite/light, but kveikja á is the everyday way to say turn on.

Because kveikja á governs the dative case.

• lampi (a lamp) is masculine; its dative singular definite form is lampanum.
  So á lampanum literally means on the lamp (i.e., activating it), with lampanum in the dative.

-num here signals dative singular + definite article for many masculine nouns (and some neuters).
Think of it as lampi + the + (dative role) → lampanum = the lamp (in dative).

Það here is a dummy/impersonal subject, like English it in it is dark. Icelandic often requires an explicit subject, so you get:

• það er myrkur = it is dark
  Even though það doesn’t refer to a specific thing.

Myrkur is commonly used as a predicative word meaning dark/darkness in the pattern það er myrkur = it is dark.
It behaves a bit like a “noun-like” adjective here (roughly darkness), and this specific construction is just the standard, natural phrasing.

Úti means outside and specifies that it’s dark outdoors.

• þegar það er myrkur = when it is dark (general)
• þegar það er myrkur úti = when it is dark outside
  You can omit úti, but then it’s less specific.

Þegar introduces a subordinate time clause (when...). In Icelandic subordinate clauses, the finite verb (here er) normally comes after the subject, unlike main-clause verb-second patterns.
So you get: þegar + subject + verb → þegar það er ...

Yes. If you front the subordinate clause, the main clause typically shows verb-second word order with inversion (the verb comes right after the fronted clause):

• Þegar það er myrkur úti, kveiki ég á lampanum.
  Notice kveiki comes before ég.

Grammatically it’s present tense, but Icelandic present tense often covers both:

• what you do now (I’m turning it on)
• what you do habitually (I turn it on / I usually turn it on)
  With þegar... it naturally reads as a habitual/general action: I turn on the lamp when it’s dark outside.

A practical guide (approximate):

• Ég: like yeh(g) (often the g is soft or barely heard)
• kveiki: roughly KVEI-ki (first syllable stressed; ei like “ay”)
• það: like tha (the ð is a voiced “th” sound, like in this)
• úti: OO-ti (long ú = “oo”)
  Main stress in Icelandic is almost always on the first syllable: KVEI-ki, LAM-pa-num, MYR-kur.

Yes, depending on what you mean:

• kveikja á lampanum = turn on the lamp (the fixture/device)
• kveikja á ljósinu = turn on the light (the light itself; very common)
  Both work; they just focus on slightly different nouns.