الصالة الرياضية وغرفة تبديل الملابس كبيرتان ونظيفتان اليوم.

Because this is a present-tense nominal sentence in Arabic. In Modern Standard Arabic, the verb to be is usually not stated in the present tense.

So:

• الصالة الرياضية وغرفة تبديل الملابس كبيرتان ونظيفتان اليوم
• literally: The gym and the changing room big and clean today
• natural English: The gym and the changing room are big and clean today

If you wanted the past, Arabic would use a form of كان:

• كانت الصالة الرياضية وغرفة تبديل الملابس كبيرتين ونظيفتين
• The gym and the changing room were big and clean

The sentence breaks down like this:

• الصالة الرياضية وغرفة تبديل الملابس = the subject
• كبيرتان ونظيفتان = the predicate
• اليوم = an adverb of time, meaning today

So the structure is:

• [subject] + [predicate] + [time expression]

More specifically, the subject is a compound subject made of two nouns joined by و:

• الصالة الرياضية = the gym / sports hall
• غرفة تبديل الملابس = the changing room

Because الرياضية is an adjective, and in Arabic adjectives normally come after the noun they describe.

So:

• الصالة = the hall
• الرياضية = sports-related / athletic

Together:

• الصالة الرياضية = the sports hall or the gym

The adjective also has الـ because Arabic adjectives must agree with the noun in several ways, including definiteness. Since الصالة is definite, الرياضية must also be definite.

This is normal adjective agreement in Arabic:

• noun + adjective
• both match in definiteness, gender, number, and case

Because غرفة تبديل الملابس is an iḍāfa phrase, often called a construct phrase.

In an iḍāfa:

• the first noun usually does not take الـ
• the whole phrase can still be definite if the final noun is definite

Here we have a chain:

• غرفة = room
• تبديل = changing
• الملابس = the clothes

So literally it is:

• room of changing the clothes

and naturally:

• the changing room

The phrase is definite because the last noun, الملابس, is definite with الـ. That definiteness extends to the whole iḍāfa phrase.

Literally, it means changing the clothes or the changing of clothes.

• تبديل is a verbal noun meaning changing, switching, or replacing
• الملابس means the clothes

So:

• غرفة تبديل الملابس
• literally: room of changing clothes
• naturally: changing room

This is a very common pattern in Arabic: a place is named by using غرفة or another noun, followed by an iḍāfa phrase that explains its function.

Because the sentence is talking about two things:

• الصالة الرياضية
• غرفة تبديل الملابس

These two nouns are joined by و, so together they form a subject referring to exactly two items. In standard Arabic, when the subject is exactly two things, the predicate often appears in the dual.

So:

• كبيرتان = two feminine things are big
• نظيفتان = two feminine things are clean

This is the same idea as:

• محمد وعلي مجتهدان
• Muhammad and Ali are hardworking

The predicate is dual because there are two subjects.

Because both nouns in the subject are grammatically feminine:

• الصالة is feminine
• غرفة is feminine

In Arabic, adjectives and predicates must agree with the nouns they describe. Since there are two nouns and both are feminine, the predicate is feminine dual:

• كبيرتان
• نظيفتان

This is grammatical gender, not biological gender. Rooms and halls are not female in meaning, but the nouns are grammatically feminine.

Here ـتان is the dual feminine nominative ending, not ordinary tanween.

So:

• كبيرتان = nominative dual feminine
• نظيفتان = nominative dual feminine

They are nominative because they are the predicate of a nominal sentence.

In different grammatical positions, the dual changes:

• nominative: ـتان
• accusative/genitive: ـتين

For example:

• كانت الصالة الرياضية وغرفة تبديل الملابس كبيرتين ونظيفتين
• after كانت, the predicate is accusative, so you get ـتين

So in your sentence, ـتان is there because the grammar requires the nominative dual.

Because here they are predicates, not adjectives directly attached to the nouns.

In Arabic, the predicate in a sentence like this is usually indefinite:

• الصالة الرياضية وغرفة تبديل الملابس كبيرتان
• The gym and the changing room are big

That is different from an adjective used inside a noun phrase. For example:

• الصالة الرياضية الكبيرة
• the big gym

There, الكبيرة is an adjective directly describing الصالة الرياضية, so it is definite.

But in your sentence, كبيرتان and نظيفتان are the statement being made about the subject, so they stay indefinite.

Yes. Both words apply to both nouns together:

• the gym is big and clean
• the changing room is big and clean

Because the two nouns are joined into one compound subject, the two predicate adjectives describe that whole subject.

If you wanted to attach an adjective directly to each noun inside the noun phrase, you would structure it differently, for example:

• الصالة الرياضية الكبيرة وغرفة تبديل الملابس النظيفة

That means something like:

• the big gym and the clean changing room

So the placement matters:

• after the whole subject = predicate of the sentence
• right after a noun = adjective modifying that noun

اليوم means today, and here it functions as an adverb of time.

It tells you when the statement is true:

• The gym and the changing room are big and clean today

Putting اليوم at the end is very natural. Arabic often places time expressions there.

It can also move, depending on emphasis, for example:

• اليوم الصالة الرياضية وغرفة تبديل الملابس كبيرتان ونظيفتان
• Today, the gym and the changing room are big and clean

But the version with اليوم at the end is straightforward and common.

Because غرفة is followed by another word in an iḍāfa construction.

A ة at the end of a word, called tāʾ marbūṭa, is often pronounced:

• like -a / -ah when the word stands alone
• like -t when the word is linked to what follows

So:

• alone: غرفة ≈ ghurfa
• in this phrase: غرفة تبديل الملابس ≈ ghurfat tabdīl al-malābis

This is very normal in Arabic and is one of the first pronunciation patterns learners notice in iḍāfa phrases.