Постоянный шум мешает мне работать.

Because мешать requires the person affected to be in the dative case, not the nominative.

So:

• я = I (subject form)
• мне = to me / for me

Russian structures this idea more like:

• Constant noise interferes with me working
• literally: Constant noise hinders to me to work

That is why мне is correct here.

Постоянный шум is in the nominative case because it is the subject of the sentence — the thing doing the action of мешает.

Breakdown:

• постоянный = constant
• шум = noise
• мешает = interferes / prevents / bothers

So постоянный шум is the thing that мешает.

Because it is the 3rd person singular present-tense form of мешать.

The subject is шум, which is:

• singular
• masculine
• 3rd person

So the verb must match that:

• мешать = to interfere, to hinder
• мешает = it interferes / it prevents

Here, шум is grammatically singular, so мешает is singular too.

Because after мешать Russian often uses this pattern:

• мешать + dative person + infinitive

Meaning:

• to prevent someone from doing something
• to interfere with someone’s doing something

So:

• мешает мне работать = prevents me from working / interferes with my working

The infinitive работать tells you what action is being hindered.

No. Russian does not need a separate word corresponding to English from in this structure.

English says:

• prevents me from working

Russian says:

• мешает мне работать

So the idea of from is built into the structure with мешать + dative + infinitive.

The stresses are:

• постоя́нный
• шум
• меша́ет
• мне
• рабо́тать

So the whole sentence is pronounced approximately as:

постоя́нный шум меша́ет мне рабо́тать

A learner should especially notice:

• постоя́нный, not по́стоянный
• меша́ет, not ме́шает
• рабо́тать, not работа́ть

Yes. Russian word order is more flexible than English word order.

The neutral order here is:

• Постоянный шум мешает мне работать.

But you could also say:

• Мне мешает работать постоянный шум.
• Работать мне мешает постоянный шум.

These all mean roughly the same thing, but the emphasis changes:

• Постоянный шум... focuses first on the noise.
• Мне мешает... emphasizes me.
• Работать мне мешает... emphasizes the activity being interrupted.

In Russian, шум is commonly used as a singular mass/collective noun meaning noise in general.

So:

• шум = noise
• шумы = noises, different sounds/noises

In this sentence, the general idea of ongoing noise is meant, so singular шум is natural.

It can overlap with all of those, depending on context.

In this sentence, мешает мне работать most naturally means:

• interferes with my work
• prevents me from working
• makes it hard for me to work

It is often stronger than just annoys me, because it suggests the noise is actually getting in the way of the action.

So Постоянный шум мешает мне работать is not just The constant noise annoys me; it is more specifically The constant noise is making it difficult/impossible for me to work.

Мешать is imperfective.

That fits well here because the sentence describes an ongoing or repeated situation:

• Constant noise interferes with my working
• Constant noise keeps me from working

If you wanted a perfective verb, you would usually use помешать, which refers more to a single completed instance of interference:

• Шум помешал мне работать. = The noise prevented me from working / The noise got in the way of my work.

So:

• мешает = is interfering, generally/continuously
• помешал = interfered once, as a completed event

Because it must agree with шум.

Шум is:

• masculine
• singular
• nominative

So the adjective must also be:

• masculine
• singular
• nominative

That gives:

• постоянный шум

If the noun changed, the adjective would change too. For example:

• постоянная проблема = a constant problem
• постоянное давление = constant pressure
• постоянные шумы = constant noises

Yes, and it is very natural, but the meaning shifts slightly.

• мешает мне работать = interferes with my working, gets in the way
• не даёт мне работать = does not let me work, more direct and stronger

So both are possible:

• Постоянный шум мешает мне работать.
• Постоянный шум не даёт мне работать.

The second one can sound a bit more forceful, as if the noise is completely stopping you.

Usually not if you want to say it interferes with me working.

Compare:

• Постоянный шум мешает мне работать. = Constant noise interferes with my working.
• Постоянный шум мешает работать. = Constant noise makes working difficult / interferes with working.

The second sentence is more general and less personal. It sounds like working in general is difficult because of the noise, not specifically for me.

So мне is important if you want to say that the speaker is the affected person.