Piątego marca mam wizytę u dentysty.

Because Polish dates use the ordinal number in the genitive case.

So piątego marca literally means something like of the fifth of March.
There is an implied word dnia (day) in the background:

• piątego marca = (piątego dnia) marca = on the fifth of March

That is why you do not use the basic dictionary form piąty here.

Examples:

• pierwszego maja = on the first of May
• dziesiątego lipca = on the tenth of July

Because month names also change case in dates.

• marzec = March
• marca = of March / in March in this date structure

After a day number, the month is normally in the genitive:

• piątego marca
• dwunastego czerwca
• trzeciego listopada

So in dates, you usually learn them like this:

• 1 marca
• 5 marca
• 20 marca

not 5 marzec.

Polish often does not need a separate preposition here.

English says:

• on the fifth of March

Polish simply says:

• piątego marca

The date expression itself already works adverbially, so no extra word like on is needed.

Yes, exactly — and that is the normal Polish way to say it.

Polish commonly uses mieć (to have) for appointments:

• Mam wizytę u dentysty. = I have a dentist appointment.
• Mam wizytę u lekarza. = I have a doctor's appointment.

This is very natural Polish. It is not awkward or too literal.

Because mam (I have) takes a direct object, and the direct object is usually in the accusative case.

• nominative: wizyta
• accusative: wizytę

So:

• mam wizytę = I have an appointment

Compare:

• To jest wizyta. = This is an appointment.
• Mam wizytę. = I have an appointment.

u dentysty means at the dentist's or with the dentist.

The preposition u is often used when talking about being at someone's place, with a professional, or under someone's care/service.

So:

• u dentysty = at the dentist's / with the dentist
• u lekarza = at the doctor's
• u fryzjera = at the hairdresser's

In this sentence, wizyta u dentysty is the standard Polish way to say a dentist appointment.

Because the preposition u requires the genitive case.

• nominative: dentysta
• genitive: dentysty

So:

• u dentysty
• u lekarza
• u nauczyciela

A useful rule:

• u + genitive

Because the whole date expression uses case agreement patterns.

In piątego marca:

• piątego is the ordinal number in the genitive
• marca is the month name in the genitive

This is just how Polish forms dates. English learners often expect only one word to change, but in Polish it is normal for multiple words in a phrase to reflect the grammar.

Yes.

• 5 marca is the usual written form
• piątego marca is how you would read it aloud

So if you see:

• Mam wizytę 5 marca.

you would normally say:

• Mam wizytę piątego marca.

Both are correct, depending on whether you are writing with digits or spelling it out.

No, Polish word order is fairly flexible.

All of these are possible:

• Piątego marca mam wizytę u dentysty.
• Mam wizytę u dentysty piątego marca.
• Mam piątego marca wizytę u dentysty.

They all mean roughly the same thing, but the emphasis changes.

The original sentence sounds very natural because it starts with the time expression:

• Piątego marca = setting the time first

That is a very common pattern.

Not in this sentence.

• wizyta u dentysty = an appointment with / at the dentist
• iść do dentysty = to go to the dentist

So:

• Mam wizytę u dentysty. = I have an appointment with the dentist.
• Idę do dentysty. = I'm going to the dentist.

do dentysty suggests movement toward the dentist, while u dentysty is the normal expression for the appointment itself.

Because some Polish nouns referring to male people end in -a but are still masculine.

dentysta is one of them, like:

• mężczyzna = man
• kolega = male friend
• artysta = artist

So even though dentysta looks a bit like a feminine noun, it declines as a masculine personal noun in important ways.

That is why the genitive is:

• dentysty

not something based on a feminine pattern.

No. In normal Polish, month names are written with a lowercase letter.

So:

• marca
• czerwca
• listopada

not:

• Marca
• Czerwca
• Listopada

This is different from English, where March must be capitalized.

A very literal breakdown is:

• Piątego marca = on the fifth of March
• mam = I have
• wizytę = an appointment / visit
• u dentysty = at the dentist's / with the dentist

So the structure is basically:

On the fifth of March, I have an appointment with the dentist.

That literal view can help explain the grammar, even if the natural English translation may be slightly different.