Learn Hardcore Russian: Шумная улица мешает мне засыпать. - The noisy street prevents me from falling asleep.

Улица is the grammatical subject of the sentence: it’s the thing that does the action (or causes the effect) expressed by the verb мешает.

In Russian, the subject is normally in the nominative case. So:

Шумная улица – subject (nominative)
мешает – verb
мне засыпать – what is being affected

Roughly: The noisy street (subject) prevents me from falling asleep (object of the prevention).

So улица must be nominative, not another case.

What exactly does мешает mean here, and how does the pattern мешать + dative + infinitive work?

The verb мешать has a few meanings:

to hinder / to prevent / to interfere with – the meaning in your sentence
to disturb / to bother
to mix / to stir (a completely different, physical meaning)

In this sentence it is meaning (1). The common construction is:

мешать кому-то делать что-то
– to hinder someone from doing something / to get in the way of someone doing something

So:

мешать – to hinder
мне – to me (dative)
засыпать – to fall asleep (infinitive)

Put together: мешать мне засыпать = to keep me from falling asleep, to make it hard for me to fall asleep.

Why is it мне (dative) and not меня (accusative) after мешает?

With the meaning to hinder / to interfere with, the verb мешать normally takes the dative case for the person who is affected:

мешать кому? – to hinder whom? (in the sense: to be in whose way?)

So:

мешать мне – to hinder me / to be in my way
мешать тебе – to hinder you
мешать ему / ей – to hinder him / her

Other similar verbs also use the dative:

помогать мне – to help me
вредить мне – to harm me

The accusative меня would be wrong here. You say:

Шумная улица мешает мне. – The noisy street bothers/gets in my way.
Шумная улица мешает мне засыпать. – The noisy street keeps me from falling asleep.

Why is it засыпать and not спать? What’s the difference?

The verbs are different:

спать – to sleep (state)
засыпать – to fall asleep (the process of beginning to sleep)

In the sentence, the problem is not sleep in general, but starting to sleep. So:

Шумная улица мешает мне засыпать.
– The noisy street keeps me from falling asleep.

If you used спать, the meaning would shift:

Шумная улица мешает мне спать.
– The noisy street keeps me from sleeping / stops me while I’m sleeping.

Both are correct, but they describe slightly different problems.

What is the difference between засыпать and заснуть?

They are an aspect pair:

засыпать – imperfective, to be falling asleep / to fall asleep (as a process or habit)
заснуть – perfective, to fall asleep (as a single completed event)

In general:

imperfective = process, repeated/habitual action, or general ability
perfective = a one-time, completed result

In your sentence, засыпать suggests a process or repeated situation:

Шумная улица мешает мне засыпать.
– The noisy street makes it hard for me to fall asleep (whenever I try; in general).

A sentence with заснуть would feel more like a specific episode, especially in the past:

Шумная улица мешала мне заснуть.
– The noisy street was preventing me from (managing to) fall asleep (that time).

So засыпать here fits a general/habitual problem very well.

Can I change the word order, like Шумная улица мне мешает засыпать or Мне шумная улица мешает засыпать?

Yes, Russian word order is flexible. All of these are grammatically correct:

Шумная улица мешает мне засыпать. – neutral, standard.
Шумная улица мне мешает засыпать. – slightly more emphasis on мне.
Мне шумная улица мешает засыпать. – stronger emphasis on мне (it is me who is affected).
Мне мешает засыпать шумная улица. – emphasis shifts to шумная улица (it is the noisy street that’s the one causing the problem).

In speech, the stress and intonation will show what is being emphasized, but the basic meaning stays the same in all those variants.

How would I negate this: how do I say The noisy street doesn’t stop me from falling asleep?

You simply add не in front of the verb мешает:

Шумная улица не мешает мне засыпать.
– The noisy street does not keep me from falling asleep.

If you want to emphasize at all, you can add an adverb:

Шумная улица совсем не мешает мне засыпать.
– The noisy street doesn’t bother my falling asleep at all.
Шумная улица нисколько не мешает мне засыпать.
– The noisy street does not interfere with my falling asleep in the least.

But the basic negation is just не before the verb.

Can I omit мне and just say Шумная улица мешает засыпать?

Yes, that is possible and natural:

Шумная улица мешает засыпать.

This sounds more general and impersonal, like:

A noisy street makes it hard to fall asleep.
A noisy street interferes with falling asleep.

There is no explicit person being mentioned; it’s about the situation in general.

If you say мешает мне засыпать, it is clearly me who is affected.

What is the nuance of шумная улица? Can I say громкая улица or оживлённая улица instead?

The adjectives are close but not identical:

шумная улица – a noisy street; lots of noise in general (cars, people, music, etc.). This is the most natural here.
громкая улица – literally loud street; this sounds odd or unnatural for a street. Громкий is more for specific sounds: громкая музыка (loud music), громкий голос (loud voice).
оживлённая улица – a busy / lively street, with many people, cars, shops. It emphasizes activity and life, not directly the noise, although busy streets are usually noisy.

In your sentence, шумная улица мешает мне засыпать is the idiomatic choice, because the problem is specifically the noise.

Could мешать here mean simply to annoy, like The noisy street annoys me when I’m trying to fall asleep?

The main meaning of мешать in this construction is to interfere with / to hinder, especially when followed by an infinitive:

мешать мне засыпать – to get in the way of my falling asleep, to prevent or complicate falling asleep.

It can imply annoyance (because something that stops you from sleeping is obviously annoying), but the focus is on the obstacle to the action, not on the feeling.

If you wanted to stress pure annoyance, you would more likely say:

Шумная улица меня раздражает. – The noisy street irritates me.
Меня раздражает шумная улица. – I’m annoyed by the noisy street.

So in your sentence, мешать is about making it hard to fall asleep, not just about feeling annoyed.

Is there another common way to say the same thing, like using не даёт instead of мешает?

Yes. A very common alternative is:

Шумная улица не даёт мне заснуть.

Here:

не даёт – does not let / does not allow
мне – to me (dative)
заснуть – to fall asleep (perfective; one-time result)

Nuance:

мешать мне засыпать – to hinder the process of falling asleep, often sounds slightly softer (it makes it hard).
не даёт мне заснуть – stronger: it doesn’t let me fall asleep at all, prevents the result.

Both are very natural; which one you choose depends on what nuance you want to highlight: general interference (мешать засыпать) or not allowing the result to happen (не давать заснуть).