Hún staðfestir tímann hjá tannlækninum strax, en ég fresta mínum tíma til næstu viku.

Yes. Tími literally means time, but very often in everyday Icelandic it also means a booked appointment or time slot, especially with doctors, dentists, hairdressers, and similar services.

So in this sentence:

• staðfesta tímann = confirm the appointment
• fresta tíma = postpone an appointment

This is very normal Icelandic usage.

Because staðfesta takes a direct object in the accusative.

The noun is tími. Its accusative singular is tíma, and with the definite article (the) it becomes tímann.

So:

• tími = an appointment / time
• tímann = the appointment

In Hún staðfestir tímann, tímann is the thing she is confirming.

Hjá often means with, at, or by, especially when talking about being with a person or at a professional’s place/service.

So:

• hjá tannlækninum = with the dentist / at the dentist’s

In English we usually say an appointment with the dentist. Icelandic commonly expresses that idea with hjá.

The preposition hjá takes the dative.

The dictionary form is tannlæknir (dentist). After hjá, it must be in the dative singular:

• tannlækni = dentist (dative singular)
• tannlækninum = the dentist (dative singular definite)

So hjá tannlækninum is grammatically:

• hjá
  • dative
• the dentist in dative form

Because fresta takes a dative object.

So after fresta, tími must be in the dative:

• tíma = appointment (dative singular)

And the possessive minn (my) has to agree with that noun in case, gender, and number:

• masculine singular dative = mínum

So:

• mínum tíma = my appointment (dative)

minn tíma would not fit here, because that is not the correct dative form.

Normally, yes. A very useful pattern to learn is:

• fresta e-u = postpone something

That e-u means the thing postponed is in the dative.

Examples:

• fresta fundinum = postpone the meeting
• fresta ferðinni = postpone the trip
• fresta mínum tíma = postpone my appointment

This is one of those verbs where learning the case together with the verb is very helpful.

Both patterns are possible in Icelandic, but they are built differently.

In this sentence, the possessive comes before the noun:

• mínum tíma

When the possessive is placed before the noun, Icelandic usually does not also use the suffixed definite article on the noun.

A different natural option would be:

• tímanum mínum

There the possessive comes after the noun, and the noun takes the definite article.

So both can work, but the structure changes.

Because til takes the genitive.

So after til, the phrase næstu viku is in the genitive:

• til næstu viku = until next week / to next week

In this sentence, it means the appointment is being moved forward so that it will happen next week.

This is a very common time expression in Icelandic. With words like næsti (next) and síðasti (last) in time phrases, Icelandic very often uses the weak adjective form:

• í næstu viku = next week
• til næstu viku = until next week
• frá síðasta ári = since last year

So even if adjective forms feel complicated at first, næstu viku is the normal expression to learn as a set phrase.

Yes. Strax means right away / immediately, and adverbs in Icelandic can often move around somewhat.

The sentence’s version is very natural:

• Hún staðfestir tímann hjá tannlækninum strax.

You may also hear:

• Hún staðfestir strax tímann hjá tannlækninum.

Both are possible. The version in your sentence sounds smooth and keeps hjá tannlækninum close to tímann.

En usually means but, and it marks a contrast.

Here the contrast is:

• She confirms the appointment right away,
• but I postpone my appointment.

So en is the natural conjunction.

Because these are different verbs, and they belong to different conjugation patterns in the present tense.

Here are the relevant forms:

• staðfesta → ég staðfesti, hún staðfestir
• fresta → ég fresta, hún frestar

So the sentence is simply using the correct present-tense forms for each subject:

• hún staðfestir = she confirms
• ég fresta = I postpone