Learn Hardcore Russian: Я нашёл нужный документ в папке на столе. - I found the necessary document in the folder on the table.

Нашёл is the active verb найти = to find (you found something).
Нашёлся is the reflexive form найтись = to be found / to turn up (the thing “got found” on its own):

Документ нашёлся. = The document turned up / was found.

What does Я нашёл tell me about the speaker (gender/number)?

In past tense, Russian verbs agree with the subject in gender (singular) and number:

я нашёл = I (male) found
я нашла = I (female) found
мы нашли = we found (no gender distinction in plural)

Why is найти perfective here? What would the imperfective be?

Найти (perfective) focuses on the result: you successfully located it.
The imperfective is находить:

Я нашёл нужный документ… = I found the right document (got the result).
Я находил нужный документ… is unusual without context; it often means I used to find / I was able to find (repeatedly), or it can appear in specific constructions (e.g., negation, general statements).

For “I was looking for it,” Russian uses a different verb: искать:

Я искал нужный документ. = I was looking for the right document.

Why is it spelled нашёл with ё? Can I write нашел?

Correct spelling is нашёл (with ё), and it affects pronunciation/stress: наш-ЁЛ.
In many texts ё is often replaced by е (нашел), but it’s still pronounced ё by native speakers. In learning materials it’s usually better to keep ё.

Why is документ in the form документ, not документа?

Because it’s the direct object of нашёл, so it’s accusative. For masculine inanimate nouns, accusative = nominative:

документ (Nom) → документ (Acc)

If it were masculine animate, accusative would match genitive (e.g., нашёл брата).

Why is нужный in that exact form?

Adjectives agree with the noun in gender, number, and case.
Документ is masculine singular accusative (inanimate), so:

нужный документ = masculine singular accusative (same ending as nominative here)

Compare:

нужную папку (feminine acc.)
нужное письмо (neuter acc.)
нужные документы (plural acc. inanimate)

What case is в папке and why?

В папке is prepositional case (also called locative in many textbooks) because в + location answers where? (где?):

в папке = in the folder (location)

If it were motion into the folder (where to? куда?), you’d use accusative:

положил документ в папку = put the document into the folder

Why is it на столе (not в столе) and what case is столе?

На is used for something being on a surface:

на столе = on the table

Столе is prepositional case of стол after на meaning location (где?).

В столе would usually mean inside the table (e.g., inside a desk/table with compartments), and it would also take prepositional for location:

в столе = in the desk/table (inside it)

How do I know whether в means “in” vs “into,” and на means “on” vs “onto”?

Russian distinguishes location vs direction mainly by case:

Location (where? где?) → prepositional: в папке, на столе
Direction (where to? куда?) → accusative: в папку, на стол

So в/на can mean in/on or into/onto depending on the case.

Why is the word order Я нашёл нужный документ в папке на столе? Can it change?

Russian word order is flexible; changes usually shift emphasis. Neutral here is:

Я нашёл нужный документ в папке на столе. (standard narrative)

Possible variations:

В папке на столе я нашёл нужный документ. (emphasizes where you found it)
Нужный документ я нашёл в папке на столе. (emphasizes it was the right document)

Does в папке на столе mean “in the folder that is on the table,” or “in the folder, on the table”?

Most naturally it means: in the folder that is on the table (the folder is located on the table; the document is inside the folder).

If you wanted to clearly separate them as two independent locations (less likely), you’d rephrase, e.g.:

Я нашёл документ в папке, которая лежала на столе. = …in the folder that was lying on the table.

What’s the difference between нужный документ and правильный документ?

нужный = needed / the one you need / required (matches your needs or purpose)
правильный = correct / right (as opposed to wrong)

In this context, нужный документ usually means the document you needed (for some task).