Ég sef betur undir sænginni þegar mér er kalt.

Because sef is the 1st person singular present form of the verb að sofa (to sleep).

The verb changes like this in the present tense:

• ég sef — I sleep
• þú sefur — you sleep
• hann/hún/það sefur — he/she/it sleeps
• við sofum — we sleep
• þið sofið — you (plural) sleep
• þeir/þær/þau sofa — they sleep

So að sofa is the infinitive, but after ég you need the conjugated form sef.

Betur means better and it is the comparative form of vel (well).

This is an irregular pattern, similar to English:

• vel — well
• betur — better
• best — best

So:

• Ég sef vel — I sleep well
• Ég sef betur — I sleep better

Since sef is a verb, Icelandic uses the adverb betur, not an adjective.

Sænginni is the definite singular form, so it means the duvet / the quilt / the covers.

The basic noun is:

• sæng — duvet, quilt

With the suffixed definite article, it becomes:

• sængin — the duvet

But after undir, the noun is in the dative here, so sængin changes to:

• sænginni — the duvet in the dative

So undir sænginni means under the duvet / under the covers.

Because the preposition undir can take either the accusative or the dative, depending on meaning.

A very common rule is:

• accusative for motion toward something
• dative for location or being in a place

Here, the sentence describes where the person sleeps, not movement:

• Ég sef undir sænginni — I sleep under the duvet

That is a static location, so Icelandic uses the dative.

Compare:

• Ég fer undir sængina — I go under the duvet
  Here there is movement, so accusative
• Ég sef undir sænginni — I sleep under the duvet
  Here there is location, so dative

Not exactly. Sæng usually means something like duvet, quilt, or sometimes bedclothes/covers depending on context.

In this sentence, undir sænginni is very natural as under the duvet or more idiomatically under the covers.

A plain blanket is more specifically:

• teppi — blanket

So sæng is more about bedding on the bed, especially the main cover you sleep under.

Because Icelandic often uses an impersonal construction for physical sensations such as being cold, hot, hungry, sleepy, and so on.

So instead of saying I am cold in the same structure as English, Icelandic says something more like:

• mér er kalt — to me is cold

Here:

• mér is the dative form of ég
• er means is
• kalt means cold

This pattern is extremely common in Icelandic.

Some similar examples:

• Mér er heitt — I am hot
• Mér er illt — I am in pain / it hurts
• Mér er sama — I don’t care / it’s all the same to me

If you said ég er kaldur/köld/kalt, that would usually describe you as a physically cold object or person in a more literal/descriptive sense, not the normal everyday way to say I feel cold.

Because this construction requires the dative case.

The pronoun ég has these relevant forms:

• ég — nominative
• mig — accusative
• mér — dative
• mín — genitive

In mér er kalt, the experiencer is put in the dative, so mér is the correct form.

This is something you often just have to learn together with the expression:

• mér er kalt
• þér er kalt
• honum er kalt
• henni er kalt

Because kalt is the neuter singular form of the adjective kaldur (cold), and that is the form commonly used in this impersonal pattern:

• mér er kalt

In these expressions, Icelandic uses the neuter singular adjective regardless of the person who feels the sensation.

So you get:

• Mér er kalt
• Henni er kalt
• Honum er kalt

Not:

• Mér er kaldur
• Henni er köld

Those would sound like normal descriptive adjective agreement, which is not how this common feeling-expression works.

Here þegar means when and introduces a subordinate clause:

• þegar mér er kalt — when I am cold

It is very commonly used for when in time clauses.

In other contexts, þegar can also mean already:

• Ég er þegar kominn — I have already arrived

So the meaning depends on context. In your sentence, because it introduces a clause with a verb, it clearly means when.

Because Icelandic does not express this idea with a nominative subject like English does. Instead, it uses the impersonal pattern mér er kalt.

So the full clause is:

• þegar mér er kalt — when I am cold

Even though English has I, Icelandic uses mér here. That is just the normal grammar of this expression.

Yes, Icelandic word order is somewhat flexible, though some versions are more natural than others.

Your sentence:

• Ég sef betur undir sænginni þegar mér er kalt.

This is a very natural neutral order.

You could also move parts around for focus, for example:

• Þegar mér er kalt, sef ég betur undir sænginni.

That means the same thing, but now the when-clause comes first. Notice that when a clause comes first, the verb in the main clause comes before the subject:

• sef ég
• not ég sef

That is the normal V2 word order pattern in Icelandic.

More specifically, it means under the duvet / under the covers, not just in bed.

If you wanted to say in bed, you would usually say:

• í rúminu — in the bed

So:

• Ég sef betur í rúminu — I sleep better in bed
• Ég sef betur undir sænginni — I sleep better under the covers

The second one emphasizes being covered up.

Yes. Ég sef betur is present tense, but like English present tense, it can describe a general truth, habit, or usual pattern.

So the sentence means something like:

• I sleep better under the covers when I’m cold
• I tend to sleep better under the duvet when I’m cold

It does not have to mean only what is happening right now.

A helpful breakdown is:

• Ég — I
• sef — sleep
• betur — better
• undir sænginni — under the duvet / under the covers
• þegar — when
• mér er kalt — I am cold

So the structure is:

• main clause: Ég sef betur undir sænginni
• subordinate clause: þegar mér er kalt

That makes it easier to see that the sentence is basically:

I sleep better under the covers + when I’m cold