Learn Hardcore Russian: Не забудь купить лук на рынке. - Don’t forget to buy onions at the market.

Не забудь (perfective, imperative) = Don’t forget (this one specific thing). It’s a single, completed “remembering” you’re being asked to ensure.
Не забывай (imperfective, imperative) = Don’t forget (in general / repeatedly), or a more “ongoing” reminder, like Keep in mind… / Don’t be forgetting….

What form is забудь? How is it built?

Забудь is the informal singular imperative of забыть (perfective: to forget).

Infinitive: забыть
Imperative (ты): забудь
Imperative (вы): забудьте The -ь is common in imperatives of some verbs and signals a “soft” ending in pronunciation.

Why does Russian use a negative imperative with не here—does it work like English don’t?

Yes, functionally it matches English don’t: не + imperative. So Не забудь… = Don’t forget….
Russian forms it directly with не plus the imperative form, not with an auxiliary like do.

Why is купить in the infinitive and not an imperative like купи?

Because the main command is Не забудь… (Don’t forget…), and what you shouldn’t forget is expressed with an infinitive:

Не забудь + infinitive = Don’t forget to + verb So купить stays infinitive: to buy.
If you wanted a direct command, you’d say Купи лук на рынке. (Buy onions at the market.)

Why is купить perfective? Could it be покупать?

Купить (perfective) is natural because it means buying as a single completed task: buy (and get it done).
Покупать (imperfective) would suggest a process/habit or repeated action (to be buying / to buy regularly), and is less typical in this specific reminder.

What case is лук here, and why doesn’t it look different?

Лук is the direct object of купить, so it’s in the accusative.
But for many inanimate masculine nouns, accusative = nominative, so it looks the same:

Nominative: лук
Accusative (inanimate): лук

Does лук mean onion or bow? How do I know?

Лук can mean both:

onion
bow (as in archery) Context decides. With купить and на рынке (buy at the market), it almost always means onion.
If you needed to clarify, you could say лук (репчатый) for onion, or лук (оружие) / лук для стрельбы for bow.

Why is it на рынке and not в рынке?

Russian commonly uses на for places understood as venues/locations of activity (markets, stations, concerts, etc.).

на рынке = at the market (normal, idiomatic) в рынке would mean literally inside the market and is generally not how you refer to the place as a shopping destination.

What case is рынке after на?

With location (not motion), на takes the prepositional case:

рынок (dictionary form)
на рынке (prepositional singular)

If it were motion to the market, you’d use accusative:

на рынок = to the market (direction)

Can the word order change? What sounds most natural?

Yes, word order is flexible. The given order is very natural:

Не забудь купить лук на рынке. Other possible orders shift emphasis:
Не забудь на рынке купить лук. (emphasizes at the market)
Лук не забудь купить на рынке. (emphasizes onions)

Is Не забудь… informal? How do I make it polite or plural?

Yes, забудь is addressed to ты (one person, informal).
Polite/plural uses вы:

Не забудьте купить лук на рынке.

Where is the stress, and how is it pronounced roughly?

Common stress:

Не забу́дь купи́ть лук на ры́нке. Notes:
забу́дь ends with a soft consonant (because of ь).
ры́нке has stress on the first syllable: РЫн-ке.