Learn Hardcore Russian: Я мою грязную обувь в ванной вечером. - I wash the dirty shoes in the bathroom in the evening.

Обувь is grammatically feminine, singular.
In the accusative singular feminine, the adjective ending -ая (грязная) typically changes to -ую (грязную).

So:

Nominative: грязная обувь (dirty shoes – as a subject)
Accusative: грязную обувь (dirty shoes – as an object)

That’s why it’s грязную, not грязный or грязная.

Why is обувь singular in Russian if it means shoes (plural) in English?

Обувь is a collective noun in Russian. It’s grammatically singular but refers to footwear in general or a set of shoes:

обувь – footwear / shoes (in general), grammatically singular
ботинки – (a pair of) shoes/boots
туфли – (a pair of) dress shoes
кроссовки – (a pair of) sneakers

In this sentence, грязную обувь means something like my dirty footwear / my dirty shoes, but grammatically it behaves as a singular feminine noun.

Why do we use в ванной and not something like в ванна?

В ванной is в + ванная in the prepositional case.

Nominative: ванная (bathroom)
Prepositional: в ванной (in the bathroom)

After в meaning in/inside, we usually use the prepositional case to show location:

в комнате – in the room
в школе – at school
в ванной – in the bathroom

So в ванна is incorrect; it must be в ванной.

Does в ванной mean “in the bath” or “in the bathroom”?

It usually means in the bathroom (short for в ванной комнате – in the bathroom).

Context decides:

в ванной (room) → in the bathroom
в ванне (tub) → in the bathtub (inside the tub itself)

So:

Я мою грязную обувь в ванной – I wash my dirty shoes in the bathroom.
Я сижу в ванне – I am sitting in the bathtub.

What case is вечером, and why is it used this way?

Вечером is the instrumental case of вечер (evening).

Russian often uses the instrumental case as an adverbial of time to mean “at/in the … (time of day)”:

утром – in the morning
днём – in/at daytime
вечером – in the evening
ночью – at night

So вечером = in the evening.

Can I change the word order? For example, say Я вечером мою грязную обувь в ванной?

Yes. Russian word order is usually flexible, and all of these are grammatically correct:

Я мою грязную обувь в ванной вечером.
Я вечером мою грязную обувь в ванной.
Вечером я мою грязную обувь в ванной.
Я мою в ванной грязную обувь вечером.

The meaning stays the same, but the emphasis changes slightly. Many speakers would naturally say:

Вечером я мою грязную обувь в ванной. (emphasis on “in the evening”)
or
Я вечером мою грязную обувь в ванной.

Your original sentence is also fine and natural.

What is the difference between я мою and я помою?

Мою – from мыть (imperfective)
Помою – from помыть (perfective)

Я мою грязную обувь.
I am washing / I wash my dirty shoes (process, habitual, or ongoing now).
Я помою грязную обувь.
I will wash my dirty shoes (future, focusing on the result – they will end up washed).

In the present tense, only imperfective (мою) is used. Помою is a future form.

Why don’t we use a reflexive verb like я моюсь here?

Мо́ться (with -сь) is reflexive and means to wash oneself:

Я моюсь в ванной вечером. – I wash (myself) in the bathroom in the evening.

In your sentence, the object is грязную обувь (dirty shoes), not the subject itself.
So we use the non‑reflexive мыть:

Я мою грязную обувь. – I wash (my) dirty shoes.
Я моюсь. – I wash myself / I’m taking a wash.

Different objects → different verbs.

How is обувь pronounced, and what does the soft sign (ь) do here?

Обувь is pronounced roughly like [Ó-boov']:

Stress on the first syllable: Ó-
The final вь is soft: the ь (soft sign) doesn’t have its own sound, but it makes the preceding consonant в soft/palatalized.

So it’s not ó-buf or ó-bus, but ó-buv' with a soft v at the end.

Is anything missing in Russian, like “my” in “my dirty shoes”?

Russian often omits possessive pronouns (мой, моя, мои, etc.) when it’s clear from context that the object belongs to the subject.

Я мою грязную обувь.
Literally: I wash dirty shoes.
Naturally understood: I wash my dirty shoes.

If you really needed to stress possession, you could say:

Я мою свою грязную обувь. – I wash my own dirty shoes (strong emphasis that they are mine).