هذه لعبة بطاقات.

Because لعبة is a feminine singular noun, and هذه is the demonstrative used for feminine singular nouns in Modern Standard Arabic.

• هذا = masculine singular this
• هذه = feminine singular this

The noun لعبة is feminine because it ends in ة (the taaʾ marbuuṭa), which is a very common sign of a feminine noun.

Also, even though بطاقات is plural, the main noun here is still لعبة. So the demonstrative agrees with لعبة, not with بطاقات.

In Arabic, the present tense of to be is usually not stated.

So a sentence like this is completely normal without any separate word for is:

• هذه = this
• لعبة بطاقات = a card game

Arabic often makes this kind of sentence with just:

• a subject/topic
• then a description or identification

So هذه لعبة بطاقات literally looks like this + card game, but it means this is a card game.

Yes. لعبة بطاقات is a noun phrase, and more specifically it is an iḍāfa (construct phrase).

That means:

• لعبة = game
• بطاقات = cards

Together, they mean something like:

• a game of cards
• naturally in English: a card game

In an iḍāfa, the first noun is the main thing, and the second noun specifies it.

So here:

• the main thing is لعبة = a game
• what kind of game? بطاقات = of cards

That is a very natural question. Arabic and English do not always build these compound ideas the same way.

English often uses a singular noun before another noun:

• card game
• chicken soup
• toothbrush

Arabic often expresses the same idea as X of Y, and the second noun may be plural:

• لعبة بطاقات = a game of cards

So even though English uses singular card in card game, Arabic can use the plural بطاقات.

Because the sentence is identifying what this is, not saying this game as a noun phrase.

Compare:

• هذه لعبة بطاقات = This is a card game
• هذه اللعبة = this game

So:

• هذه لعبة بطاقات is a full sentence
• هذه اللعبة is usually a noun phrase meaning this game

This is an important distinction:

• demonstrative + indefinite predicate often gives This is a ...
• demonstrative + definite noun often gives this ...

When ة (taaʾ marbuuṭa) is at the end of a word by itself, it is often pronounced like -a or -ah in pause.

But when the word is connected grammatically to what follows, especially in an iḍāfa, the ة is pronounced as t.

So here:

• لعبة by itself: roughly luʿbah
• لعبة بطاقات: luʿbat biṭāqāt...

That is because لعبة is the first part of an iḍāfa, so the ة is sounded as t.

In fully vocalized formal Arabic, it would be:

هَذِهِ لُعْبَةُ بِطاقاتٍ

A few useful points:

• هَذِهِ = hādhihi
• لُعْبَةُ = luʿbatu
• بِطاقاتٍ = biṭāqātin

Why these endings?

• لعبةُ is nominative because it is the predicate of the sentence
• بطاقاتٍ is genitive because it is the second noun in an iḍāfa
• لعبة does not take tanwīn here, because the first noun of an iḍāfa does not take tanwīn

So you get لعبةُ بطاقاتٍ, not لعبةٌ بطاقاتٍ.

No. You do not need هي here.

هذه already means this, and that is enough to start the sentence.

So the normal simple sentence is:

هذه لعبة بطاقات

You may sometimes see هذه هي ... in other contexts for emphasis or clarity, but it is not necessary here.

Because the whole phrase is indefinite: a card game, not the card game.

In an iḍāfa, the definiteness of the whole phrase depends mainly on the second noun.

So:

• لعبة بطاقات = a card game / a game of cards
• لعبة البطاقات = the card game or the game of the cards, depending on context

Also remember: in an iḍāfa, the first noun normally does not take ال.

Normally, no. In this exact form, it is understood as a sentence: This is a card game.

If you want this card game as a noun phrase, Arabic would normally make the noun definite, for example:

• هذه اللعبة = this game

If you want to say this card game, you would usually need a more explicit definite structure, depending on context.

So for a learner, the safest reading of هذه لعبة بطاقات is:

• هذه = subject/topic
• لعبة بطاقات = predicate

That gives the sentence meaning This is a card game.