Eftir að hafa heimsótt fósturheimilið ákváðum við að ættleiða hvolpinn.

It means after visiting or, more literally, after having visited.

• eftir = after
• að hafa heimsótt = to have visited

So the sentence begins with a completed action that happened before the deciding.

Hafa is used to show that the visiting happened before the main action.

In other words:

• first: they visited
• then: they decided

So að hafa heimsótt gives the idea having visited or after having visited.

Because after hafa, Icelandic uses the form that corresponds to English visited in have visited.

• heimsækja = to visit
• hafa heimsótt = have visited

So heimsótt is the correct form after hafa here.

This is because Icelandic normally follows a verb-second pattern in main clauses.

When the sentence starts with something other than the subject, such as:

Eftir að hafa heimsótt fósturheimilið

the finite verb comes next:

ákváðum

and then the subject:

við

So:

• Við ákváðum ... = We decided ...
• Eftir að hafa heimsótt ..., ákváðum við ... = After visiting ..., we decided ...

This is very normal Icelandic word order.

Ákváðum is the past tense, first person plural form of ákveða.

So it means:

we decided

You do not need a separate word for we inside the verb, but Icelandic still often includes the pronoun við.

Here að is the infinitive marker, like to in English.

• ættleiða = adopt
• að ættleiða = to adopt

After ákveða, Icelandic commonly uses að + infinitive:

• ákveða að gera eitthvað = decide to do something

So ákváðum við að ættleiða hvolpinn means we decided to adopt the puppy.

They are similar, but they appear in different structures.

• eftir að hafa heimsótt = after having visited
• ákveða að ættleiða = decide to adopt

In both places, að is connected to an infinitive-type construction, but the surrounding grammar is different.

The -ið is the attached definite article, meaning the.

The base noun is:

• fósturheimili = foster home

So:

• fósturheimilið = the foster home

It is the object of heimsótt, so it is in the accusative, but for this neuter noun the nominative and accusative singular look the same.

Because it is the definite accusative singular form of hvolpur.

• hvolpur = puppy
• hvolpinn = the puppy

It is in the accusative because it is the direct object of ættleiða.

A useful comparison:

• hvolpurinn = the puppy as subject
• hvolpinn = the puppy as object

The understood subject is the same as the main clause subject: við = we.

So the sentence means:

After we had visited the foster home, we decided to adopt the puppy.

Icelandic often leaves that subject unstated in this kind of non-finite construction when it is clear from the main clause.

Yes. A full clause is possible.

For example:

• Eftir að við heimsóttum fósturheimilið, ákváðum við að ættleiða hvolpinn.
• Eftir að við höfðum heimsótt fósturheimilið, ákváðum við að ættleiða hvolpinn.

The version with að hafa heimsótt is a more compact way to express the same general idea, especially when the subject is the same in both parts.

Yes. Ættleiða can be used for adopting animals as well.

So with hvolpinn, it naturally means to adopt the puppy. In everyday usage, this is a very normal choice of verb.