Je cherche ma trousse, car j’ai besoin d’un crayon et d’une gomme pour écrire.

In this sentence, ma trousse means my pencil case (the small case where you keep pens, pencils, erasers, etc.).

The word trousse on its own can mean different kinds of kits, depending on context:

• une trousse d’école – a school pencil case
• une trousse de maquillage – a make-up bag
• une trousse de secours – a first-aid kit

So in a school / writing context, trousse is naturally understood as a pencil case.

Because trousse is a feminine noun in French.

• ma = my (before a feminine singular noun)
• mon = my (before a masculine singular noun or a vowel sound)
• mes = my (before any plural noun)

So:

• ma trousse (feminine)
• mon crayon (masculine)
• mes trousses (plural)

In French, chercher already includes the idea of “for”. You do not add another preposition:

• Je cherche ma trousse. = I’m looking for my pencil case.
  (not: ✗ je cherche pour ma trousse)

Some examples:

• Je cherche mon téléphone. – I’m looking for my phone.
• Elle cherche un travail. – She is looking for a job.

Yes, you can say both:

• Je cherche ma trousse, car j’ai besoin d’un crayon…
• Je cherche ma trousse parce que j’ai besoin d’un crayon…

Main differences:

• car is a bit more formal / written, often used in explanations and in writing.
• parce que is more common in everyday speech.

Meaning-wise, both introduce a reason (“because”). In most spoken situations, people would naturally say parce que.

With avoir besoin, you must always use de before what you need.

Pattern:

• avoir besoin de + noun
• avoir besoin de + verb (infinitive)

So:

• J’ai besoin d’un crayon. (correct)
• ✗ J’ai besoin un crayon. (incorrect – missing de)

More examples:

• J’ai besoin de temps. – I need time.
• J’ai besoin de dormir. – I need to sleep.

d’un is just the contracted form of de + un:

• de + un → d’un
• de + une → d’une

In front of un/une, de almost always contracts like this:

• J’ai besoin d’un crayon.
• Elle parle d’une amie.

The same thing happens with le and les:

• de + le → du
• de + les → des

In careful French, you should repeat de / d’ before each noun:

• d’un crayon et d’une gomme (recommended)

Saying:

• ✗ d’un crayon et une gomme
  sounds incomplete or incorrect to most speakers, because avoir besoin de is felt to apply directly to each object.

So:

• J’ai besoin d’un crayon et d’une gomme. – I need a pencil and an eraser.

Because:

• crayon is masculine → un crayon
• gomme is feminine → une gomme

There is no simple rule that always tells you the gender; you mostly have to learn the noun together with its article:

• un crayon (a pencil)
• une gomme (an eraser)

Dictionaries normally show the gender:

• crayon (n. m.) – masculin
• gomme (n. f.) – féminin

Yes, but it would change the meaning slightly.

• un crayon et une gomme = a pencil and an eraser (not specific ones; any will do)
• le crayon et la gomme = the pencil and the eraser (the listener knows which ones you mean, or you’re talking about them in a specific context)

In your sentence, you just need some pencil and eraser to write, so un / une is more natural.

pour + infinitive expresses a purpose: “in order to do X”.

• pour écrire – (in order) to write
• pour étudier – (in order) to study
• pour comprendre – (in order) to understand

In your sentence:

• … j’ai besoin d’un crayon et d’une gomme pour écrire.
  = I need a pencil and an eraser to write / in order to write.

You could, but it would add extra information or feel a bit unnatural if nothing follows.

• pour écrire avec un crayon – to write with a pencil
• pour écrire quelque chose – to write something

In your sentence, the idea is just the general purpose: to write, so simple pour écrire is the most natural and idiomatic.

In spoken and written French, je + ai always contracts to j’ai. This is called elision.

• je ai → ✗ (never used)
• j’ai → ✓ (correct form)

This happens when je is followed by a verb starting with a vowel sound:

• j’aime (je aime) – I like
• j’habite (je habite) – I live
• j’ai (je ai) – I have

In French, it’s very common to put a comma before car when it introduces a reason.

• Je ne viens pas, car je suis malade.
• Je cherche ma trousse, car j’ai besoin d’un crayon…

In everyday writing, you’ll almost always see that comma. In very short sentences, some people might omit it, but putting it is considered standard and correct.

Yes, both are possible, with the same basic meaning:

• J’ai besoin d’un crayon et d’une gomme.
• Il me faut un crayon et une gomme.

Both mean “I need a pencil and an eraser.”
Nuance:

• J’ai besoin de… is more neutral/common.
• Il me faut… can sound a bit more direct or insistent, like “I really need / I must have…”.