Learn Hardcore Russian: В этом кафе всегда шумно вечером. - It is always noisy in this café in the evening.

After the preposition в meaning in / at (location), the noun phrase must be in the prepositional case.
The demonstrative adjective этот (this) changes in the prepositional case to этом:

этот кафе → в этом кафе

So это кафе is the basic this café (nominative), and в этом кафе is in this café (prepositional).

What case is кафе in, if the form doesn’t change?

Кафе is an indeclinable neuter noun: its form is the same in all cases.
In this sentence it is logically in the prepositional case, because it follows в with a location meaning (in / at this café).
You can’t see the case on the noun itself, so you determine it by the preposition and the role in the sentence: в этом кафе = prepositional.

Why is it шумно and not шумный?

Шумно is the short predicative form of the adjective шумный (noisy).
Russian uses these short forms to describe a state with no explicit subject, similar to it is noisy:

В этом кафе всегда шумно. = It’s always noisy in this café.

If you said шумный, you would be making it an attribute of a noun, e.g. шумное кафе = a noisy café (adjective before a noun), which is a different structure.

Is шумно an adverb or an adjective?

In school grammar, шумно here is usually called a short-form adjective or a category of state word.
It does not modify a verb (like a typical adverb would) and does not agree with any noun; instead, it describes the overall situation:

Шумно. = It is noisy.

So grammatically it’s closer to a special predicative adjective form than to a normal adverb.

Why is there no verb есть (is) in the Russian sentence?

In Russian, the present-tense form of быть (to be) is normally omitted in sentences like this.
So instead of a literal В этом кафе всегда есть шумно, Russian simply says В этом кафе всегда шумно.
Adding есть here would sound unnatural or overly emphatic; the basic neutral statement uses no explicit to be in the present.

Could we also say В этом кафе всегда бывает шумно вечером? What is the difference?

Yes, В этом кафе всегда бывает шумно вечером is grammatically correct.
Бывает (from быть) emphasizes that this is a repeated situation that happens from time to time or regularly, not literally every single minute.
Without бывает, В этом кафе всегда шумно вечером sounds more like a general constant characteristic: evenings there are (practically) always noisy.

What case is вечером, and why is that case used for time?

Вечером is the instrumental case of вечер (evening).
The instrumental case is often used to express a time when something happens, especially with times of day:

утром (in the morning)
днём (in the daytime)
вечером (in the evening)
ночью (at night)

So вечером here means in the evening / in the evenings (depending on context).

What is the difference between вечером and по вечерам?

вечером = in the evening; can refer to a specific evening or to evenings in general, depending on context.
по вечерам = in the evenings / on evenings (in general); clearly indicates a regular, repeated action or situation.

In your sentence, В этом кафе всегда шумно вечером already implies a habitual situation.
If you say В этом кафе всегда шумно по вечерам, you make the idea of regularity even stronger and more explicit.

Can the word order change, for example to Вечером в этом кафе всегда шумно? Does the meaning change?

Yes, Russian word order is quite flexible, and several variants are possible:

В этом кафе всегда шумно вечером. (neutral: place → frequency → state → time)
Вечером в этом кафе всегда шумно. (emphasis on evenings; you start with the time)
В этом кафе вечером всегда шумно. (slight emphasis on evenings within this café)

The core meaning stays the same; changing the order mainly affects which part is highlighted or sounds more prominent in the sentence.

Why is в used (в этом кафе) and not на?

The preposition в is used for being inside or within an enclosed space: в кафе, в доме, в магазине.
На is more typical for surfaces, open areas, or certain idiomatic locations: на столе, на улице, на работе, на вокзале.
A café is seen as a place you go inside, so Russian naturally uses в кафе, not на кафе.

Does this sentence have a subject? Is it an impersonal sentence?

There is no explicit grammatical subject in В этом кафе всегда шумно вечером.
The sentence describes a state (шумно) that simply exists in a certain place and time.
This is a typical impersonal sentence (безличное предложение) in Russian, where the predicate does not refer to any specific subject like he, she, or it.

Does шумно agree with кафе in gender or number?

No, шумно here does not agree with кафе.
It is a fixed neuter short-form used to describe a state in an impersonal sentence, not an adjective agreeing with a noun.
So you use шумно in this pattern regardless of the noun:

В лесу шумно. (It is noisy in the forest.)
В доме шумно. (It is noisy in the house.)
На улице шумно. (It is noisy in the street.)

Could we say something like В этом кафе всегда шумит вечером instead of шумно?

Not in that exact form. Шумит is a verb (to make noise, to be noisy) and normally needs a subject:

Вечером в этом кафе шумят гости. = The guests make noise in this café in the evenings.

If you say В этом кафе всегда шумно вечером, you avoid naming a subject and just describe the overall atmosphere.
Using шумно is the natural way to say it is noisy there.

What is the stress pattern in this sentence?

The main stresses are:

В Э́том кафе́ всегда́ шу́мно ве́чером.

Word by word:

э́том – stress on the first syllable
кафе́ – stress on the second syllable
всегда́ – stress on the second syllable
шу́мно – stress on the first syllable
ве́чером – stress on the first syllable