دکتری که نزدیک خانه ما است، امروز وقت دارد.

That final ی is the indefinite marker. It often corresponds to a/an in English.

So دکتری literally looks like a doctor.

However, when Persian uses a noun plus a relative clause, English may translate it as either a doctor who... or the doctor who..., depending on context. So even though دکتری is formally indefinite, the whole phrase can still refer to a fairly specific person in practice.

که introduces a relative clause. In this sentence, it means something like who, that, or which.

So:

• دکتری که نزدیک خانه ما است
  = the/a doctor who is near our house

A useful thing to remember is that Persian uses که for people and things alike. English separates them into who, that, and which, but Persian usually just uses که.

Because the relative clause needs a verb: is.

The clause is literally structured like:

• که نزدیک خانه ما است
• who near our house is

Persian usually puts the verb at the end of the clause, so است comes last.

In more natural spoken Persian, this often becomes:

• دکتری که نزدیک خانه ماست...
• or دکتری که نزدیک خانه ما هست...

خانه ما means our house.

Persian possession is usually built as:

• possessed noun + possessor

So:

• خانهٔ ما = house of us = our house

The little linking sound called ezafe is often present in pronunciation, but it is often not written in ordinary text. So خانه ما is commonly written, even though the pronunciation may be closer to خونه‌ٔ ما or خانهٔ ما depending on style.

نزدیک means near or close to, and in Persian it can be followed directly by a noun phrase.

So:

• نزدیک خانه ما = near our house

You do not need a separate word equivalent to English to here.

In careful pronunciation, you may hear an ezafe-like linking sound:

• نزدیکِ خانهٔ ما

But in ordinary writing, it is very common to write simply:

• نزدیک خانه ما

Yes, literally وقت دارد means has time.

But in real usage, it often means:

• is free
• is available
• has an opening
• has time to see someone

With a doctor, it often suggests that the doctor is available today, or has an appointment slot today.

So the phrase is natural even though the literal wording is has time.

امروز means today, and Persian word order is fairly flexible with time expressions.

In this sentence:

• دکتری که نزدیک خانه ما است، امروز وقت دارد.

امروز comes before the final verb phrase, which is very natural.

You could also hear other arrangements, such as:

• امروز دکتری که نزدیک خانه ما است وقت دارد.

That puts a bit more emphasis on today.

So the position of امروز is normal, but not the only possible one.

Yes.

Both است and هست can mean is, but they differ in style:

• است is more formal and more common in writing
• هست is common in speech and less formal writing

So these are both possible:

• دکتری که نزدیک خانه ما است...
• دکتری که نزدیک خانه ما هست...

In very natural spoken Persian, you may also hear the shorter attached form:

• دکتری که نزدیک خونهٔ ماست...

In isolation, دکتری can sometimes mean doctorate or PhD.

But in this sentence, it clearly means doctor because of the context:

• near our house
• has time today

That sounds like a person, not an academic degree.

So here, دکتری is understood as a doctor.

Because the subject is singular:

• دکتری = a doctor

So the verb is singular too:

• وقت دارد = has time

If the subject were plural, the verb would change:

• دکترهایی که نزدیک خانه ما هستند، امروز وقت دارند.
• The doctors who are near our house have time today.

Also, Persian verbs agree with person and number, not with grammatical gender.

Not really.

Many Persian writers would write the sentence without a comma:

• دکتری که نزدیک خانه ما است امروز وقت دارد.

The comma here mainly shows a pause. Persian punctuation is sometimes less rigid than English punctuation in relative clauses.

Also, in English, a clause like who is near our house would normally be restrictive, so English usually would not use a comma there. Persian writers may still insert one for readability or rhythm.

A natural spoken version would often be:

• دکتری که نزدیک خونهٔ ماست، امروز وقت داره.

Changes you can notice:

• خانه → خونه in casual speech
• است → ماست as the attached spoken form
• دارد → داره in colloquial speech

So the written sentence is correct and normal, but everyday speech often sounds softer and more contracted.