Learn Hardcore Russian: Этот неприятный шум мешает мне спать. - This unpleasant noise keeps me from sleeping.

кому? = dative = мне, тебе, ему, ей, нам, вам, им
So мешает мне спать literally means (it) interferes with me sleeping / prevents me from sleeping. Accusative (меня) would sound wrong with this verb in this structure.

What is the basic grammar structure of the whole sentence?

It’s:

Этот неприятный шум = subject (nominative)
мешает = verb (3rd person singular, present tense)
мне = indirect object (dative)
спать = infinitive (what action is being hindered)

So the subject is the noise, and the verb agrees with шум.

Why is мешает singular?

Because the subject шум is singular masculine. Russian verb agreement is with the subject:

шум мешает (singular)
If it were plural: шумы мешают.

Why does этот look like Этот here—what does it agree with?

Этот is the masculine nominative singular form of этот (this). It agrees with шум (masculine singular nominative):

этот шум (m)
эта музыка (f)
это окно (n)
эти звуки (pl)

It’s capitalized (Этот) only because it’s the first word of the sentence.

Why is the adjective неприятный in that form?

Adjectives agree with the noun in gender, number, and case.
шум is masculine, singular, nominative → неприятный (masc. sg. nom.).
Compare:

неприятная новость (fem. sg. nom.)
неприятное чувство (neut. sg. nom.)
неприятные звуки (pl. nom.)

Is the word order fixed? Could I rearrange it?

It’s fairly flexible, but changes emphasis. Common options:

Этот неприятный шум мешает мне спать. (neutral)
Мне мешает спать этот неприятный шум. (emphasis on me / the problem for me)
Спать мне мешает этот неприятный шум. (emphasis on sleeping)

Russian uses word order more for focus than for basic grammar roles (cases carry a lot of that information).

Why is спать an infinitive and not something like сплю?

Because the construction is мешать + кому? + infinitive: to hinder someone from doing something.
Using сплю would create a different structure and would sound unnatural here. Russian doesn’t usually say the equivalent of The noise interferes that I sleep; it uses the infinitive: мешает (мне) спать.

Can I drop мне? What would change?

Yes: Этот неприятный шум мешает спать.
That becomes more general/impersonal: This unpleasant noise makes it hard to sleep (in general / for people here).
With мне, it clearly means it prevents me from sleeping.

Is мешать imperfective or perfective, and does it matter here?

мешать is imperfective. In the present tense, Russian normally uses the imperfective anyway. It expresses an ongoing/repeated situation: the noise is (currently) interfering.

There is a perfective counterpart помешать, which typically refers to a single completed interference event and is used in past/future contexts, e.g. Шум помешал мне уснуть (The noise prevented me from falling asleep).

How is this different from не даёт мне спать?

Both can mean prevents me from sleeping, but the feel differs:

мешает мне спать = bothers / interferes / makes it hard to sleep (can be somewhat softer)
не даёт мне спать = doesn’t let me sleep (often stronger, more direct)

What’s the difference between мешает мне спать and мешает мне уснуть?

мешает мне спать focuses on the process/state of sleeping: it interferes with my sleeping (I can’t sleep well / can’t stay asleep).
мешает мне уснуть focuses on falling asleep (a one-time transition): it prevents me from falling asleep.

Where is the stress, and how would you pronounce it roughly?

Common stress: Э́тот неприя́тный шум меша́ет мне спа́ть.
Notes:

меша́ет has stress on -а-.
спа́ть has stress on а.
неприя́тный has stress on -я́-.