Dziesiątego lutego mam wizytę u lekarza o siódmej rano.

In Polish, dates usually do not need a preposition.
So dziesiątego lutego naturally means on the tenth of February.

English says on February 10th, but Polish often just uses the date by itself:

• Spotkanie jest piątego marca. = The meeting is on March 5th.

They are in the genitive case, which is very common in Polish date expressions.

• dziesiątego = of the tenth
• lutego = of February

You can think of it as something like on the tenth day of February.

Other examples:

• pierwszego stycznia = on the first of January
• trzeciego maja = on the third of May

Yes. Mam wizytę is a very natural Polish way to say I have an appointment.

Polish often uses mieć (to have) the same way English does here:

• Mam spotkanie. = I have a meeting.
• Mam egzamin jutro. = I have an exam tomorrow.

So mam wizytę u lekarza is a standard, everyday phrase.

Because it is the direct object of mam, and mieć takes the accusative case.

The dictionary form is:

• wizyta = appointment / visit

But after mam, it changes to:

• mam wizytę

This is a very common feminine noun pattern:

• mam książkę
• mam kawę
• mam wizytę

Because wizyta u lekarza means an appointment with / at the doctor’s. That is the normal idiomatic phrase.

• u lekarza focuses on being with that person / at that person’s place or in their care
• do lekarza usually goes with a verb of motion, like I’m going to the doctor

So:

• Mam wizytę u lekarza. = I have an appointment with the doctor.
• Idę do lekarza. = I’m going to the doctor.

You would normally not say mam wizytę do lekarza.

Lekarza is in the genitive case because the preposition u requires the genitive.

• dictionary form: lekarz
• after u: u lekarza

More examples:

• u mamy = at mum’s / with mum
• u dentysty = at the dentist’s
• u kolegi = at a male friend’s place

When Polish gives a clock time with o (at), it normally uses the locative form of an ordinal-style hour expression:

• o pierwszej = at one
• o drugiej = at two
• o siódmej = at seven

Historically, it is like saying at the seventh hour.
That is why you get siódmej, not just siedem.

Because o is the normal Polish preposition for at with specific clock times.

So:

• o siódmej = at seven
• o ósmej trzydzieści = at 8:30
• o dziewiątej wieczorem = at nine in the evening

This is different from broader time expressions, where Polish may use other structures, for example:

• w lutym = in February
• rano = in the morning
• w poniedziałek = on Monday

Because o siódmej by itself can mean at seven, but it does not automatically tell you whether that is morning or evening.
Adding rano makes it clear that it is 7 a.m.

• o siódmej rano = at seven in the morning
• o siódmej wieczorem = at seven in the evening

Sometimes context is enough, but rano removes ambiguity.

Because Polish usually leaves out subject pronouns when they are not needed.

The verb form mam already tells us the subject is I:

• mam = I have

So Ja mam wizytę... is possible, but it sounds more emphatic, like: I have an appointment...
In a neutral sentence, just mam wizytę... is more natural.

In Polish, names of months are normally written with a lowercase letter.

So:

• luty
• marzec
• listopad

This is different from English, where February and March are capitalized.

Yes. Polish word order is fairly flexible, and the meaning usually stays the same, though the emphasis can change.

Your sentence:

• Dziesiątego lutego mam wizytę u lekarza o siódmej rano.

A very natural alternative:

• Mam wizytę u lekarza dziesiątego lutego o siódmej rano.

Starting with Dziesiątego lutego puts the date first, which can sound like you are highlighting when the appointment is.