هذه بطاقة لعب.

Because بطاقة is a feminine singular noun, and Arabic demonstratives agree with the noun in gender and number.

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

So with بطاقة, you use هذه.

In Arabic, the verb to be is usually not stated in the present tense when you make a simple sentence like this.

So:

• هذه بطاقة لعب = literally this a playing card
• natural English: This is a playing card

This kind of sentence is called a nominal sentence.

If you wanted was or will be, Arabic would normally use a verb.

Arabic does not have a separate word for the indefinite article a/an.

Instead, indefiniteness is usually shown by the noun simply not having الـ.

So بطاقة here is understood as a card, not the card.

In fully vowelled formal Arabic, indefinite nouns often take tanwīn. But here there is an extra detail: بطاقة is the first part of a construct phrase, so it does not take tanwīn.

No, لعب here is not an adjective. It is a noun.

The phrase بطاقة لعب is a common Arabic structure called iḍāfa (construct phrase):

• بطاقة = card
• لعب = play / game

So the whole phrase literally works like card of play, which English naturally expresses as playing card.

Arabic often uses noun + noun structures where English might use an adjective or a compound noun.

Because بطاقة is the first noun in an iḍāfa.

In an iḍāfa:

• the first noun does not take الـ
• the first noun does not take tanwīn

So in full formal pronunciation, this would be:

• بِطاقةُ لَعِبٍ
• biṭāqatu laʿibin

not biṭāqatun laʿibin.

This is a very common pattern in Arabic.

Because the phrase is indefinite: a playing card, not the playing card.

In an iḍāfa, the second noun controls whether the whole phrase is definite or indefinite.

• بطاقةُ لعبٍ = a playing card / an indefinite phrase
• بطاقةُ اللعبِ = a definite phrase, something more like the card of the game/play

So leaving لعب without الـ helps keep the whole expression indefinite.

That final ة is tāʾ marbūṭa.

When the word stands alone or comes at a pause, it is usually pronounced like -a or -ah:

• بطاقة → biṭāqa / biṭāqah

But when the word is followed grammatically by another word in a construct phrase, the t sound appears:

• بطاقة لعب → biṭāqat laʿib

So in this sentence, the t is heard because بطاقة is connected to لعب in an iḍāfa.

In careful Modern Standard Arabic, هذه is pronounced hādhihi.

At the end of a pause, many speakers shorten it a bit, so you may hear something closer to hādhih.

For a learner reading MSA aloud, hādhihi is the safest pronunciation.

A full formal reading would be:

hādhihi biṭāqatu laʿibin

In Arabic with short vowels, that would be:

هٰذِهِ بِطَاقَةُ لَعِبٍ

The important thing to notice is:

• بطاقةُ is nominative because it is the predicate of the sentence
• لعبٍ is genitive because it is the second noun in the iḍāfa

In normal everyday writing, these short vowels are usually omitted, so you just see:

هذه بطاقة لعب

It is a complete sentence: This is a playing card.

Structure:

• هذه = the subject-like element, this
• بطاقة لعب = the predicate, telling you what it is

Compare:

• هذه بطاقة لعب = This is a playing card
• هذه البطاقة = this card

So adding the indefinite phrase بطاقة لعب makes it a full statement rather than just a noun phrase.