Мне нужна салфетка, потому что у меня насморк.

Both are possible in Russian, but they are used differently.

Мне нужна салфетка is the most natural everyday way to say I need a tissue/napkin.
It literally works more like To me, a tissue is necessary.

Я нуждаюсь в салфетке is grammatically correct, but it sounds much more formal, bookish, or dramatic. Russian usually prefers the кому нужен/нужна/нужно/нужны + noun pattern for ordinary needs.

So in daily speech:

• Мне нужна салфетка. = I need a tissue.
• Я нуждаюсь в помощи. = I am in need of help.
  This sounds more formal and is more common with abstract nouns like help, support, care, etc.

Because this construction uses the dative case for the person who needs something.

The pattern is:

кому? нужен/нужна/нужно/нужны + что?

So:

• мне нужен... = I need...
• тебе нужен... = you need...
• ему нужна... = he needs...

Here, мне is the dative form of я.

A good way to think about it is:

• я = I
• мне = to me / for me

So Мне нужна салфетка is structurally closer to A tissue is needed for me than to I need a tissue.

Because нужен agrees with the thing that is needed.

The noun салфетка is:

• feminine
• singular

So the correct form is нужна.

Agreement forms:

• нужен — masculine singular
• нужна — feminine singular
• нужно — neuter singular
• нужны — plural

Examples:

• Мне нужен телефон. — I need a phone.
• Мне нужна салфетка. — I need a tissue/napkin.
• Мне нужно время. — I need time.
• Мне нужны деньги. — I need money.

It is in the nominative case.

That can surprise English speakers, because in English need takes a direct object. But in Russian, this construction does not work like a normal transitive verb.

In Мне нужна салфетка:

• мне = dative
• салфетка = nominative

The noun stays in nominative because it is the thing that is needed, and нужна agrees with it.

Салфетка can mean different kinds of paper or cloth wipes depending on context:

• napkin
• tissue
• sometimes serviette

In this sentence, because of у меня насморк (I have a runny nose), a learner would naturally understand салфетка as something used to wipe your nose — often a tissue.

In everyday Russian, people may also say:

• бумажная салфетка — paper napkin / paper tissue
• носовой платок — handkerchief

So салфетка is a broad word, and context tells you which English word fits best.

Russian usually does not use a verb equivalent to English have in the same way.

Instead, possession is commonly expressed with:

у + genitive + есть / zero in the present tense

So:

• у меня = at me / by me
• насморк = a runny nose / nasal congestion

Literally, the structure is something like:

At me, there is a runny nose.

But naturally it means:

I have a runny nose.

This у меня... structure is extremely common in Russian:

• У меня есть брат. — I have a brother.
• У меня машина. — I have a car.
• У меня насморк. — I have a runny nose.

Because у requires the genitive case.

The pronoun changes like this:

• я — nominative
• меня — genitive/accusative
• мне — dative

So:

• у меня = at me / I have
• для меня = for me
• ко мне = toward me

In this sentence, у меня is the normal possession/condition pattern.

In the present tense, Russian often omits есть in simple possession statements.

So both of these are possible in principle:

• У меня есть брат.
• У меня брат.

But with conditions, illnesses, and similar states, Russian very often just uses the noun without есть:

• У меня температура. — I have a fever.
• У меня кашель. — I have a cough.
• У меня насморк. — I have a runny nose.

Adding есть here is usually unnecessary and may sound unnatural in many contexts.

Насморк specifically means a runny nose or nasal congestion.

It does not exactly mean the whole illness a cold.

So:

• У меня насморк. = I have a runny nose.
• У меня простуда. = I have a cold.

Sometimes in real life the meanings overlap in context, because a runny nose often comes with a cold, but the words are not the same.

Because потому что introduces a subordinate clause, and Russian normally separates it with a comma.

So:

• Мне нужна салфетка, потому что у меня насморк.

This is standard punctuation, just like English often uses a comma before because when the sentence is written this way.

Yes. Russian word order is more flexible than English word order, because case endings show grammatical roles.

The neutral order here is:

Мне нужна салфетка, потому что у меня насморк.

But other orders are possible for emphasis:

• Салфетка мне нужна, потому что у меня насморк.
  Emphasizes салфетка.
• Потому что у меня насморк, мне нужна салфетка.
  Starts with the reason.
• У меня насморк, поэтому мне нужна салфетка.
  Uses therefore / so instead.

Even though the order can move, the original sentence is the most straightforward and natural for a learner.

No, not in standard Russian.

With the нужен/нужна/нужно/нужны construction, the thing needed is normally in the nominative, not the accusative.

So the correct form is:

• Мне нужна салфетка.

Not:

• Мне нужно салфетку.

The mistake happens because English speakers expect the noun after need to be a direct object. But Russian is using a different structure here.

The form of нужен changes to match the noun.

Examples:

• Мне нужен платок. — I need a handkerchief.
  (платок = masculine)
• Мне нужно полотенце. — I need a towel.
  (полотенце = neuter)
• Мне нужны салфетки. — I need tissues/napkins.
  (салфетки = plural)

So in your sentence:

• салфетка is feminine singular
• therefore: нужна

A learner-friendly pronunciation guide would be roughly:

Мне нужна салфетка, потому что у меня насморк.
mneh noozh-NAH sal-FYET-ka, pa-ta-MOO shta oo mi-NYA NAS-mark

A few useful notes:

• нужна — stress on the last syllable: нужна́
• салфетка — stress on фет: салфе́тка
• потому что is often pronounced more like патаму шта in normal speech
• меня — stress on the last syllable: меня́
• насморк — stress on the first syllable: на́сморк

If you want to sound natural, потому что is one of those phrases where everyday pronunciation is often more reduced than the spelling suggests.