غرفة الانتظار مثل هذه الغرفة.

Because this is a present-tense nominal sentence. In MSA Arabic, the verb to be is usually omitted in the present.

So:

• غرفة الانتظار مثل هذه الغرفة
• literally: the waiting room like this room
• naturally in English: The waiting room is like this room

If you wanted was, Arabic would use a verb such as كان.

It is a nominal sentence.

Its two main parts are:

• غرفة الانتظار = the subject/topic
• مثل هذه الغرفة = the predicate/comment

So the structure is basically:

• The waiting room
  • is like this room

This is one of the most common sentence patterns in Arabic.

It is an iḍāfa construction, often understood as X of Y.

So:

• غرفة = room
• الانتظار = waiting

Together:

• غرفة الانتظار
• literally: room of waiting
• natural English: waiting room

This is a very common way Arabic builds compound nouns.

Because in an iḍāfa, the first noun normally does not take الـ.

So Arabic says:

• غرفة الانتظار

not:

• الغرفة الانتظار

The whole phrase is still definite because the second noun الانتظار is definite. So غرفة الانتظار means the waiting room, not just a waiting room.

Because the first noun in an iḍāfa loses tanwīn.

So you get:

• غرفةُ الانتظارِ

not:

• غرفةٌ الانتظارِ

This is a standard rule of the construction.

Because غرفة is a feminine noun.

Arabic demonstratives agree with the gender of the noun:

• هذا = this, masculine
• هذه = this, feminine

So:

• هذه الغرفة = this room

In MSA, when a demonstrative directly modifies a noun, the noun is usually definite with الـ.

So:

• هذه الغرفة = this room

But:

• هذه غرفة

usually works as a full sentence meaning this is a room, not the noun phrase this room.

That is why هذه الغرفة is the normal form here.

مثل means like or similar to.

So:

• مثل هذه الغرفة = like this room

For a learner, it is fine to treat مثل much like English like. In traditional grammar, it is originally a noun with the sense of likeness or similarity, but the practical meaning here is simply like.

A careful fully vocalized version would be:

غُرْفَةُ الاِنْتِظارِ مِثْلُ هذِهِ الغُرْفَةِ

Main points:

• غرفةُ is nominative because it is the subject of the nominal sentence
• الانتظارِ is genitive because it is the second term of the iḍāfa
• مثلُ is nominative because it is the predicate
• الغرفةِ is genitive in مثل هذه الغرفة

In normal Arabic writing, these short vowels are usually omitted.

Because غرفة ends in ة, called tāʾ marbūṭa.

Its pronunciation changes by context:

• in pause, it is usually pronounced -ah
• when connected, the t sound appears

So:

• غرفة by itself at the end of speech sounds like ghurfah
• غرفة الانتظار sounds like ghurfat al-intiẓār or, with full endings, ghurfatu l-intiẓār

That is why the same written word can sound slightly different.

A careful reading with case endings would be:

ghurfatu l-intiẓāri mithlu hādhihi l-ghurfati

In normal pause-based reading, you will often hear something closer to:

ghurfat al-intiẓār mithl hādhihi l-ghurfah

Both reflect the same sentence; the difference is mostly about whether full case endings are being pronounced.

Yes, you can, but in simple present-tense MSA it is usually not necessary.

The basic, most natural pattern is:

• غرفة الانتظار مثل هذه الغرفة

Adding هي can add emphasis or clarity in some contexts, but the version without it is the normal straightforward sentence.