Learn Hardcore Russian: Мне нужно минимум пять минут тишины вечером. - I need at least five minutes of silence in the evening.

Because Russian often expresses need with an impersonal construction: кому? + нужно/надо.
So мне is the dative case meaning to me / for me (the person who experiences the need).
Literal structure: To me (it is) necessary...

What exactly is нужно here? Is it a verb?

Нужно is a predicative word (often taught like an impersonal “verb-like” word) meaning necessary / needed.
It doesn’t change for person (no I need / you need forms). Instead you change the person with the dative: мне / тебе / ему / нам etc.

Why is нужно in neuter singular, not matching anything?

In impersonal sentences, нужно commonly appears in the default neuter singular form.
There is no grammatical “subject” like я here for it to agree with.

Compare:

Мне нужно пять минут. (impersonal; default neuter)
Мне нужна тишина. (personal; тишина is feminine, so нужна)

What’s the difference between Мне нужно... and Мне надо...?

They’re very close.

надо is often a bit more conversational and can sound like have to / need to.
нужно can sound slightly more neutral or “necessary,” but in everyday speech they often interchange.

Your sentence would also work as: Мне надо минимум пять минут тишины вечером.

Why is it минимум пять минут, not минимум пяти минут?

Both are possible, with a style difference:

минимум пять минут is very common in speech and feels more direct.
минимум пяти минут is more “grammatical/bookish,” treating минимум more like a noun that governs the genitive.

So you may hear both; your version is natural.

Why is it пять минут (not пять минуты), and what case is минут?

After numbers 5 and higher, Russian uses the genitive plural:

пять минут (genitive plural of минута)
шесть часов, десять дней, etc.

Numbers 2–4 behave differently (e.g., две минуты, три минуты).

Why is тишины in the genitive too? What does пять минут тишины mean structurally?

This is a very common “amount of something” pattern: [amount] + [noun in genitive] meaning X of Y.

пять минут тишины = five minutes of silence

So:

минут = genitive plural because of пять
тишины = genitive singular because it’s the “substance/quality” measured (silence)

Similar examples:

стакан воды = a glass of water
кусок хлеба = a piece of bread

Why is вечером in that form? What case is it?

Вечером is the instrumental case form of вечер, used adverbially to mean in the evening.
Russian often uses instrumental like this for time expressions:

утром (in the morning)
днём (in the daytime)
вечером (in the evening)

Can I also say в вечер or в вечере for “in the evening”?

Not in this meaning. For “in the evening (as a time of day),” the natural choices are:

вечером (most common)
по вечерам (in the evenings, regularly)

в вечер isn’t used for “in the evening” in standard Russian.

Is the word order fixed? Where can вечером go?

It’s flexible. All of these can work, with slightly different emphasis:

Мне нужно минимум пять минут тишины вечером. (neutral)
Вечером мне нужно минимум пять минут тишины. (sets time frame first)
Мне вечером нужно минимум пять минут тишины. (emphasizes evening as the relevant time)

Could I replace тишины with тишина?

It depends on what you mean:

пять минут тишины = “five minutes of silence” (a measurable amount)
мне нужна тишина = “I need silence” (silence as the thing you need in general)

So in your sentence, тишины fits the “X minutes of Y” idea best.

Does this sentence sound natural in everyday Russian, or is it too formal?

It sounds natural. The only thing that can slightly change the tone is your choice of минимум:

минимум пять минут is common and conversational.
You could also say хотя бы пять минут тишины = “at least five minutes of silence,” which feels even more spoken and emotional.