Eftir skráningu nota ég bókasafnskortið í sjálfsafgreiðsluvélinni.

Nota is the present tense, first-person singular form of the verb að nota (to use).

So:

• að nota = to use
• ég nota = I use

In this sentence, it matches the subject ég.

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

The sentence starts with the time phrase Eftir skráningu (After registration). When something other than the subject comes first, the finite verb usually stays in second position, so the subject moves after it:

• Eftir skráningu nota ég ...

If you started with the subject instead, you would get:

• Ég nota bókasafnskortið ... eftir skráningu

Both are possible, but the original sentence puts emphasis on when the action happens.

Yes, normally yes. Icelandic usually does not drop subject pronouns the way some other languages do.

Even though nota tells you it is first person singular, a full sentence normally still includes ég:

• ég nota

Leaving out ég would usually sound incomplete or marked unless the context is very special.

Because the preposition eftir (after) takes the dative case in this meaning.

The basic form is:

• skráning = registration

After eftir, it becomes:

• eftir skráningu

So the ending shows the case required by the preposition.

Yes, skráningu is a noun: registration.

Icelandic often uses a noun phrase after a preposition where English might use a gerund like registering. So:

• eftir skráningu can correspond to after registration
• and in many contexts also to after registering

This is a very common pattern in Icelandic.

Icelandic very often forms compound nouns as a single word.

Here:

• bókasafn = library
• kort = card
• bókasafnskort = library card

Then Icelandic usually adds the definite article onto the end of the noun, instead of using a separate word like English the:

• bókasafnskortið = the library card

So there is no separate word for the here because it is built into the noun.

The verb nota takes the accusative case, so bókasafnskortið is the direct object here.

However, the noun kort is neuter, and in the singular definite, the nominative and accusative forms are the same:

• nominative: kortið
• accusative: kortið

So even though it is an object, its form does not visibly change.

Because the preposition í can take two different cases:

• dative for location: in / inside / at
• accusative for motion into

Here the meaning is location or place of use: the card is used in/at the self-service machine, not moved into it. So Icelandic uses the dative:

• í sjálfsafgreiðsluvélinni

It is a compound noun plus the definite article and case ending.

The main part is:

• sjálfsafgreiðsluvél = self-service machine

This can be broken down roughly as:

• sjálfs- = self-
• afgreiðsla = service / checkout / handling
• vél = machine

Because the whole phrase is after í in a location meaning, it is in the dative singular definite, so sjálfsafgreiðsluvél becomes:

• sjálfsafgreiðsluvélinni = in the self-service machine / at the self-checkout machine