Learn Hardcore Russian: Повесь пальто в шкаф, чтобы в комнате был порядок. - Hang the coat in the wardrobe so that the room is tidy.

Повесь is the imperative (command) form of the verb повесить (perfective aspect), meaning hang (it up) (and finish the action).
It’s addressed to ты (singular informal), but ты is usually omitted because the imperative already implies it.

Why is it повесить/повесь (perfective) and not вешать/вешай (imperfective)?

Russian often uses the perfective imperative when you want someone to do a single, completed action: Hang it up (once), put it away.
Вешай пальто... (imperfective) sounds more like be in the process of hanging / do it habitually / start doing it, depending on context. For a one-time instruction, повесь is the natural choice.

What does the soft sign ь in повесь do? Is it pronounced?

The ь is not a separate sound; it marks that the preceding consonant is soft (palatalized): -сь is pronounced like a softened s sound.
It’s also part of the standard imperative ending for some verbs.

Why is пальто not changing its ending? What case is it here?

Пальто is an indeclinable noun in Russian (it doesn’t change form by case).
In this sentence it functions as the direct object, so logically it’s accusative, but the form looks the same as nominative.

Is пальто masculine, feminine, or neuter? How do I know if it doesn’t decline?

Пальто is neuter. Many indeclinable nouns ending in -о are treated as neuter.
You see the gender in agreement elsewhere, e.g. новое пальто (neuter adjective).

Why is it в шкаф and not в шкафу?

Because в + accusative is used for motion/direction (into): put it into the wardrobe → в шкаф.
В шкафу is location (in the wardrobe, where something already is), using в + prepositional.

Does шкаф mean a wardrobe, a closet, or a cabinet?

Шкаф is a general word for a wardrobe/closet/cabinet, depending on context.
If you specifically mean a clothes wardrobe, шкаф is still common; you can also hear шкаф для одежды.

Why is there a comma before чтобы?

Because чтобы... introduces a subordinate clause (a purpose/result clause). In Russian, subordinate clauses are normally separated by a comma:
Повесь пальто в шкаф, чтобы...

Why does the clause use был (past tense) instead of something like present tense?

After чтобы, Russian typically uses the past tense form to express a desired or intended outcome:
чтобы ... был/была/было ≈ so that ... would be.
Even though it looks like past tense, it’s functioning like a subjunctive/desired result construction here.

Why is it был порядок and not было порядок?

Because порядок is masculine, and был agrees with it in gender and number:

порядок (masc.) → был
тишина (fem.) → была тишина
чисто (adverb/impersonal) would avoid this agreement altogether.

What does в комнате был порядок literally mean in Russian structure?

Literally it’s like In the room there was order. Russian often expresses tidiness with the noun порядок rather than an adjective like tidy.
So the idea is: make it so that order exists in the room.

How would I make it polite or address more than one person?

Use the plural/polite imperative: Повесьте пальто в шкаф, чтобы в комнате был порядок.

Повесь = informal singular (ты)
Повесьте = polite singular (вы) or plural (you all)