Ég á það til að skilja vasaklútinn eftir heima þegar ég flýti mér út.

It is an idiomatic expression meaning I tend to..., I have a habit of..., or I’m liable to....

So here Ég á það til að... does not mean literal ownership. You should learn eiga það til að + infinitive as one chunk.

Because eiga það til að is a fixed expression.

• það here is not referring to a specific thing.
• til is part of the idiom.
• að introduces the infinitive that follows.

So it is best not to translate it word by word. Just recognize the whole pattern eiga það til að + verb = to tend to + verb.

No. On its own, skilja often does mean understand, but here it is part of the verb skilja eftir, which means leave behind.

That is very common in Icelandic: a verb can change meaning when combined with a particle such as eftir, upp, út, and so on.

Because skilja eftir is a verb + particle combination, and in Icelandic the object often comes between the verb and the particle.

So:

• skilja eftir = leave behind
• skilja vasaklútinn eftir = leave the handkerchief behind

This split is normal and natural.

Because vasaklútinn is the definite singular accusative form: the handkerchief.

Here it is the direct object of skilja eftir, so it appears in the accusative.

Basic forms are:

• vasaklútur = a handkerchief, nominative
• vasaklútinn = the handkerchief, accusative definite

The ending -inn is the suffixed definite article, roughly like English the.

Because heima means at home, while heim means home / to home in the sense of direction.

Here the idea is that the handkerchief gets left at home, so a location word is needed:

• heima = at home
• heim = homeward, to home

So skilja ... eftir heima is the correct choice.

Because the verb is flýta sér, a reflexive verb meaning to hurry. This verb takes a dative reflexive pronoun.

So with ég, you get:

• ég flýti mér = I hurry
• þú flýtir þér
• hann / hún flýtir sér

So mér is correct because the verb requires that form.

Út means out or outwards.

So:

• flýta mér = hurry
• flýta mér út = hurry out / rush out

It adds the idea of moving outward, leaving the house or building.

Literally, þegar means when. But because the main clause describes a repeated habit, English often naturally understands it as when(ever).

So the sense is: this tends to happen when / whenever I rush out.

Icelandic still simply uses þegar here.

Yes. If the subordinate clause comes first, the main clause shows normal Icelandic inversion:

• Þegar ég flýti mér út, á ég það til að skilja vasaklútinn eftir heima.

In the original sentence, the main clause starts with Ég, so you get:

• Ég á það til að...

This is a very common Icelandic word-order pattern.