Przez hałas trudno mi zasypiać.

In this sentence, przez means because of or due to, not the more literal through or across.

So:

• przez hałas = because of the noise

This is a very common Polish use of przez, especially when talking about the cause of a problem, often with a slightly negative sense:

• Spóźniłem się przez korki. = I was late because of traffic.
• Nie mogę pracować przez ten hałas. = I can’t work because of this noise.

Przez takes the accusative case. The noun hałas is masculine inanimate, and in the singular, its accusative form is the same as its nominative form.

So:

• nominative: hałas
• accusative: hałas

That is why you see:

• przez hałas

and not some visibly different form.

Mi means to me and is the dative form of ja.

In this pattern, Polish often uses a dative pronoun to show the person who experiences the difficulty:

• trudno mi = it is hard for me
• łatwo mi = it is easy for me

So:

• Przez hałas trudno mi zasypiać. = Because of the noise, it’s hard for me to fall asleep.

You could replace mi with another person:

• trudno mu = hard for him
• trudno jej = hard for her
• trudno nam = hard for us

Polish very often uses an impersonal structure here.

• trudno = difficult / hard
• mi = to me

So trudno mi zasypiać literally works like:

• it is hard for me to fall asleep

This is more natural in Polish than trying to say something like I am difficult or I have difficulty in the same way English might.

A fuller version could be:

• Jest mi trudno zasypiać.

But in everyday Polish, jest is often omitted, so trudno mi zasypiać sounds very normal.

Because trudno here is not an adjective describing a noun. It is a predicative adverb-like form used in impersonal sentences.

Compare:

• To jest trudne zadanie. = This is a difficult task.
  Here you use an adjective: trudne, because it describes zadanie.

But:

• Trudno mi pracować. = It’s hard for me to work.
  Here trudno is used as a general statement about difficulty.

This pattern is very common:

• łatwo mi = it’s easy for me
• trudno mi = it’s hard for me
• przyjemnie mi = it’s pleasant for me

Because zasypiać means to fall asleep or to be getting to sleep, while spać means to sleep.

That is an important difference:

• zasypiać = the process of beginning sleep
• spać = already being asleep

So the sentence is about difficulty falling asleep, not difficulty sleeping in general.

Compare:

• Trudno mi zasypiać. = It’s hard for me to fall asleep.
• Źle śpię. = I sleep badly.
• Nie mogę spać. = I can’t sleep.

This is an aspect question.

• zasypiać = imperfective
• zasnąć = perfective

The imperfective zasypiać is used here because the sentence expresses a general, repeated, or ongoing difficulty:

• Przez hałas trudno mi zasypiać. = Because of the noise, falling asleep is hard for me / I have trouble falling asleep because of the noise.

If you used the perfective zasnąć, it would usually sound more like a single completed event:

• Przez hałas trudno mi zasnąć. = Because of the noise, it’s hard for me to fall asleep (this time / in this situation)

Both can be possible, but:

• zasypiać = more general or habitual
• zasnąć = more one-time, result-focused

Yes, usually.

These are close, but they are not exactly the same:

• Przez hałas trudno mi zasypiać.
  = Because of the noise, I find it hard to fall asleep.
  This sounds a bit more general, descriptive, and less absolute.

• Nie mogę zasnąć przez hałas.
  = I can’t fall asleep because of the noise.
  This sounds stronger and often more immediate.

So if you want to express difficulty, trudno mi zasypiać is very natural.
If you want to say I’m unable to fall asleep, nie mogę zasnąć is often the better choice.

No, Polish word order is fairly flexible. The sentence you have is natural, but other orders are also possible.

For example:

• Przez hałas trudno mi zasypiać.
• Trudno mi zasypiać przez hałas.
• Mi trudno zasypiać przez hałas. — possible, but more marked

The difference is usually about emphasis rather than basic meaning.

• Przez hałas... puts the cause first: Because of the noise...
• Trudno mi... starts with the difficulty itself: It’s hard for me...

The version you were given is very natural because it highlights the reason first.

Yes. A fuller version would be:

• Przez hałas jest mi trudno zasypiać.

But in normal Polish, especially in everyday speech, jest is often omitted in this kind of sentence:

• Przez hałas trudno mi zasypiać.

Both are correct, but the shorter version is more natural and common.

Normally, in this sentence, mi is the natural choice.

• trudno mi zasypiać = the standard, neutral phrasing

Mnie is the stronger form and is used mainly for emphasis or contrast:

• Mnie trudno zasypiać, ale jemu nie. = I have trouble falling asleep, but he doesn’t.

So:

• mi = neutral
• mnie = emphasized or contrastive

Very often, yes. When przez means because of, it commonly introduces something seen as a cause of trouble, inconvenience, or some unwanted result.

Examples:

• przez deszcz = because of the rain
• przez chorobę = because of illness
• przez hałas = because of the noise

For more neutral because of, Polish often uses other expressions too, such as:

• z powodu
• z powodu hałasu = because of the noise

So in your sentence, przez hałas sounds very natural because noise is an annoying cause.

A native speaker would probably feel it as:

• Przez hałas = because of the noise
• trudno mi = it’s hard for me
• zasypiać = to fall asleep

So the overall pattern is:

cause + difficulty + experiencer + infinitive

This pattern is very useful in Polish. You can build many similar sentences:

• Przez stres trudno mi się skupić. = Because of stress, it’s hard for me to concentrate.
• Bez okularów trudno mi czytać. = Without glasses, it’s hard for me to read.
• Po kawie trudno mi zasnąć. = After coffee, it’s hard for me to fall asleep.

It is a very natural and common Polish way to talk about difficulty.