Mam już umówioną wizytę u lekarza na piątek.

Because umówioną agrees with wizytę.

• wizyta is a feminine singular noun
• in this sentence it is in the accusative: wizytę
• the adjective/participle describing it must match in:
  • gender
  • number
  • case

So:

• nominative: umówiona wizyta
• accusative: mam umówioną wizytę

That is why you see -ą here.

Because it is the direct object of mam.

The basic structure is:

• mam wizytę = I have an appointment

Since mieć usually takes a direct object, wizyta changes to the accusative:

• nominative: wizyta
• accusative: wizytę

Then umówioną matches that accusative form.

This is a very common Polish pattern:

• mam
  • adjective/participle + noun

Literally, it is something like:

• I have an appointment that is arranged

But in natural English it means:

• I already have an appointment booked/arranged

So Polish often expresses the idea of a completed arrangement as a state you now have, rather than focusing only on the action of booking it.

Compare:

• Umówiłem wizytę. = I booked the appointment.
  Focus: the action.
• Mam umówioną wizytę. = I have an appointment booked.
  Focus: the result/state.

Już means already.

So:

• Mam umówioną wizytę. = I have an appointment booked.
• Mam już umówioną wizytę. = I already have an appointment booked.

It suggests that the arrangement has been made before now, often with a nuance like:

• it is taken care of
• there is no need to arrange it now
• this step has already been completed

Because wizyta u lekarza is the normal Polish way to say a visit to/with the doctor.

The preposition u often means:

• at someone's place
• with a person, especially a professional such as:
  • u lekarza = at the doctor's / with the doctor
  • u dentysty = at the dentist's
  • u fryzjera = at the hairdresser's

By contrast:

• do lekarza usually expresses movement toward the doctor:
  • Idę do lekarza. = I’m going to the doctor.

So:

• wizyta u lekarza = the appointment/visit itself
• iść do lekarza = to go to the doctor

Because the preposition u requires the genitive case.

The dictionary form is:

• lekarz = doctor

After u, it becomes:

• u lekarza

This is a standard case pattern:

• u lekarza
• u dentysty
• u nauczyciela
• u mamy

Because na piątek is commonly used when something is scheduled for Friday.

In this sentence, the idea is:

• the appointment has been arranged for Friday

So na fits the idea of setting or booking something for a specific date.

Compare:

• Mam wizytę na piątek. = I have an appointment for Friday.
• Mam wizytę w piątek. = I have an appointment on Friday.

Both can make sense, but the nuance is a little different:

• na piątek emphasizes the scheduling/assignment of the appointment
• w piątek emphasizes when it takes place

With appointments, na piątek is especially natural.

Yes.

It comes from the verb:

• umówić = to arrange, to schedule, to make an appointment

The form umówiona is like a passive/adjectival participle meaning:

• arranged
• booked
• scheduled

So:

• umówiona wizyta = an arranged/booked appointment

Polish uses these participle forms very naturally where English might prefer a past participle adjective like booked or scheduled.

Yes, absolutely.

That means roughly:

• I already booked an appointment with the doctor for Friday.

It is correct, but the emphasis is slightly different.

• Umówiłem już wizytę...
  Focuses on the action: I booked it
• Mam już umówioną wizytę...
  Focuses on the current result: I have it booked

Both are natural. The version in your sentence sounds especially natural when you want to say that the matter is already settled.

Yes. Polish word order is flexible, although some orders sound more natural in certain contexts.

The neutral version is:

• Mam już umówioną wizytę u lekarza na piątek.

You could also hear:

• Już mam umówioną wizytę u lekarza na piątek.
• Na piątek mam już umówioną wizytę u lekarza.
• U lekarza mam już umówioną wizytę na piątek.

The meaning stays basically the same, but the focus shifts:

• putting już earlier can stress already
• putting na piątek earlier can stress Friday
• putting u lekarza earlier can stress with the doctor

For learners, the original order is a very good default.

It can mean both, depending on context.

• wizyta literally means visit
• in many contexts, especially medical ones, it also means appointment

So:

• wizyta u lekarza can mean:
  • the actual visit to the doctor
  • the booked doctor’s appointment

In this sentence, because of mam już umówioną, the meaning is clearly appointment.

Polish has no articles like English a/an/the.

So wizytę can mean:

• a visit/appointment
• the visit/appointment

The exact meaning comes from context.

In this sentence, English naturally uses:

• I already have an appointment with the doctor for Friday

But Polish does not need a separate word for an.

Yes, very easily. This pattern is extremely common.

For example:

• Mam już umówioną wizytę u dentysty na środę.
  = I already have a dentist appointment for Wednesday.
• Mam umówioną wizytę u okulisty.
  = I have an appointment with the eye doctor.
• Mam wizytę u fryzjera na jutro.
  = I have a hair appointment for tomorrow.

So wizyta u + genitive is a very useful structure to learn.

No. The same idea works with many time expressions when something is arranged or assigned for a certain time.

For example:

• umówić coś na jutro = to arrange something for tomorrow
• umówić coś na poniedziałek = to arrange something for Monday
• umówić coś na wieczór = to arrange something for the evening
• umówić coś na trzecią = to arrange something for three o’clock

So na is very common with appointments, meetings, and scheduled events.