Kérek egy időpontot az orvoshoz holnapra.

Because kérek is the indefinite (alanyi) conjugation: you’re asking for an unspecified thing: egy időpontot (an appointment).
Kérem is the definite (tárgyas) conjugation, used when the object is definite (e.g. kérem az időpontot = I’m asking for the appointment—a specific one already identified).

-t marks the accusative case (direct object). The verb kér (to ask/request) normally takes an object in the accusative:

• kérek egy kávét = I’d like a coffee
• kérek egy időpontot = I’d like an appointment

Not strictly. You can say:

• Kérek időpontot az orvoshoz holnapra. (very natural, slightly more “generic”)
• Kérek egy időpontot… (also natural; emphasizes one appointment slot)

Hungarian often drops egy when it’s clearly “one” from context.

Hungarian typically uses case endings instead of prepositions.
orvos + -hoz = orvoshoz means to (someone), to the place of (someone), often “to the doctor’s / to see the doctor.”
So az orvoshoz functions like “to/with the doctor” in the sense of booking a visit.

• az orvoshoz = to the (specific) doctor (the one relevant in the situation: your GP, that clinic’s doctor, etc.)
• orvoshoz (no article) can sound more general or “as a service”: to a doctor / to see a doctor, and is also common in fast, practical speech.

If you mean “to a (not yet identified) doctor,” you can also say egy orvoshoz, but that’s less common in this appointment-booking context.

Hungarian has two forms of “the”:

• a before consonant sounds
• az before vowel sounds

Since orvos starts with a vowel sound (o-), it becomes az orvos.

holnap = tomorrow
holnapra = for tomorrow / by tomorrow (as the scheduled date)

The ending -ra/-re often marks a target date/time or “for (that day)” in scheduling:

• hétfőre = for Monday
• jövő hétre = for next week
• holnapra = for tomorrow

Hungarian word order is flexible and reflects emphasis. All of these can be natural:

• Kérek egy időpontot az orvoshoz holnapra. (neutral)
• Holnapra kérek egy időpontot az orvoshoz. (emphasizes tomorrow)
• Az orvoshoz kérek egy időpontot holnapra. (emphasizes to the doctor)

The meaning stays basically the same; the focus shifts.

Kérek… is commonly polite in service/request situations (reception, phone calls, etc.).
Even more polite/softer options:

• Kérnék egy időpontot… (conditional = “I would like…”, very common)
• Szeretnék kérni egy időpontot… (“I’d like to ask for…”, slightly longer/formal)

Yes—időpont is the standard everyday word for an appointment time slot (especially at doctors, offices, etc.). You’ll often hear:

• Időpontot kérek. / Időpontot szeretnék kérni. = I’d like an appointment.
• Van szabad időpont? = Is there a free slot?

A few key points:

• kérek: long é (like “ay” but Hungarian), stress on the first syllable: KÉ-rek
• idő in időpont: ő is a long rounded vowel (like German ö, but long)
• orvoshoz: the h is pronounced; -hoz has a short o
• holnapra: stress on HOL-, and -ra is a short, clear ra