Learn Icelandic: Þessi gamli kóngur sefur.

The word þessi (this) is a demonstrative pronoun, which triggers the weak form of the adjective. Because kóngur is a masculine noun in the nominative case, the weak adjective gets the ending -i, making it gamli.

Shouldn't kóngur have the definite article attached to it, like kóngurinn?

No, and this is a common mix-up! When you use the demonstrative pronoun þessi (this) in Icelandic, the noun usually stays in its indefinite form. So you say Þessi kóngur (This king) rather than Þessi kóngurinn.

The infinitive of the verb is sofa (to sleep). Why does the vowel change to e in sefur?

This is a common vowel shift for certain strong verbs in Icelandic. When conjugated in the present tense singular (I, you, he/she/it), the stem vowel o shifts to e. That is why að sofa becomes kóngurinn sefur (the king sleeps).

What gender is þessi in this sentence?

It is masculine. The word þessi is actually used for both masculine and feminine nominative forms of "this", but here it is specifically matching the masculine noun kóngur. If the noun were neuter, we would use þetta instead.