Окна моей квартиры выходят на тихий двор.

Because окна means windows, not window.

In Russian, the verb agrees with the subject in number:

• окно выходит = the window faces / looks out onto
• окна выходят = the windows face / look out onto

So выходят is plural because the subject окна is plural.

Because after a noun like окна (windows), Russian usually puts the owner in the genitive case.

So:

• моя квартира = my apartment (subject form, nominative)
• окна моей квартиры = the windows of my apartment

This is very common in Russian:

• дверь дома = the door of the house
• крыша машины = the roof of the car
• окна моей квартиры = the windows of my apartment

English often uses of or the possessive 's. Russian usually uses the genitive instead.

Because моей has to agree with квартиры, not with окна.

Here квартиры is:

• feminine
• singular
• genitive

So the possessive мой changes to match that:

• nominative feminine: моя
• genitive feminine: моей

That is why you get:

• моя квартира but
• окна моей квартиры

Here it is singular genitive, meaning of the apartment.

This can be confusing because квартиры can also be nominative plural:

• квартиры = apartments
• моей квартиры = of my apartment

The word моей makes it clear that this is genitive singular, because моей cannot go with nominative plural квартиры.

That is a very common question. Выходить can indeed mean to go out / to exit, but with things like windows, doors, or balconies, it often means:

• to face
• to overlook
• to open onto
• to look out onto

So in this sentence, окна выходят на двор means the windows face the yard/courtyard.

This is a normal and very common Russian usage:

• Окна выходят на улицу. = The windows face the street.
• Балкон выходит на море. = The balcony overlooks the sea.

Because after выходить на, Russian uses на + accusative to show what the windows face onto.

So:

• выходить на двор
• выходить на улицу
• выходить на площадь

Here двор is masculine animate? No, двор is masculine inanimate, so its accusative singular is the same as the nominative:

• nominative: тихий двор
• accusative: тихий двор

That is why the form does not visibly change.

By contrast, на тихом дворе would be prepositional, and it would mean something like in/on the quiet yard, which is not the structure used here.

Because выходить на is the normal pattern for saying that something faces or opens onto a place.

Think of it as a set expression:

• окна выходят на двор
• дверь выходит на балкон
• дом выходит на реку

You should learn выходить на + accusative as a chunk when talking about location/orientation.

Yes. Russian word order is flexible, though the original order is neutral and natural:

• Окна моей квартиры выходят на тихий двор.

Other orders are possible for emphasis:

• На тихий двор выходят окна моей квартиры.
• Окна выходят на тихий двор.
  (if моей квартиры is already understood from context)

Changing the order usually changes focus or style, not the core meaning.

Yes, that is possible in some contexts, and native speakers do say things like:

• окна смотрят во двор
• окна выходят во двор
• окна выходят на двор

But выходят на двор is especially standard and neutral when describing what the windows face onto.

A rough comparison:

• выходят на двор = face onto the courtyard
• смотрят во двор = look into the courtyard

The exact nuance can vary a little, but all are understandable.

Because Russian has no articles like English a and the.

So квартира can mean:

• an apartment
• the apartment
• sometimes just apartment in a general sense

The same is true for двор:

• a courtyard
• the courtyard

The context tells you how definite it is. In this sentence, English would usually use the windows of my apartment and a/the quiet courtyard, but Russian does not mark that with separate words.

Тихий is just the adjective modifying двор.

It must agree with двор in:

• gender: masculine
• number: singular
• case: accusative

Since двор is masculine singular inanimate, the accusative looks the same as the nominative, so you get:

• тихий двор

Compare:

• тихая улица = a quiet street
• тихое море = a quiet sea
• тихий двор = a quiet courtyard

So this is a straightforward adjective-noun agreement pattern.