Ég skola vaskaklútinn og heng hann upp svo hann þorni.

-inn is the definite article suffix in Icelandic (roughly the).
vaskaklútur (dishcloth/washcloth) is masculine, and in the accusative singular definite it becomes vaskaklútinn = the washcloth.

It’s accusative singular because it’s the direct object of the verb skola (to rinse). Many basic transitive verbs take an accusative object, and skola is one of them.

In Icelandic, pronouns follow grammatical gender, not natural gender.
vaskaklútur is masculine, so it’s referred to with hann (and later again with hann).

upp is a common verb particle/adverb used with movement/placement verbs. hengja upp means hang up (not just hang).
Word order:

• heng hann upp is very natural when the object is a pronoun.
• With a full noun, you’ll often also see heng upp vaskaklútinn, but pronouns strongly prefer sitting before the particle: heng hann upp.

Because the sentence has two verbs joined by og (and) with the same subject. Icelandic commonly omits repeating the subject in the second clause:

• Ég skola … og heng … = I rinse … and (I) hang …

You can repeat ég, but it often sounds slightly more emphatic or deliberate.

Both are present tense, 1st person singular:

• ég skola = I rinse / I am rinsing
• ég heng = I hang / I am hanging

Like English, Icelandic present can cover both habitual and “right now,” depending on context.

svo here means so that / in order that, introducing a purpose/result clause:

• … svo hann þorni = … so that it dries

It’s a very common way to express purpose with a following verb form that often differs from the “plain” present.

þornar is the normal present indicative (it dries / is drying).
þorni is the present subjunctive form, which is commonly used in clauses like svo … (so that …) to express intended outcome/purpose:

• svo hann þorni = so that it (will) dry

In everyday Icelandic, you may sometimes hear indicative after svo, but þorni is the more “textbook”/traditional choice here.

It’s from þorna = to dry (intransitive: to become dry).
So hann þorni means it dries / it becomes dry (in the subjunctive here because of svo).

Yes. vaskaklútur is a compound:

• vaska- relates to washing/dishes (from vaskur in the sense of washing up)
• klútur = cloth/rag

So literally it’s something like washing-up cloth.