Learn Hardcore Russian: Учитель заметил ошибку в моём докладе и попросил меня исправить её. - The teacher noticed a mistake in my report and asked me to correct it.

Russian past tense is formed from the infinitive stem + -л (historically a past marker).
заметить → заметил (masculine), заметила (feminine), заметило (neuter), заметили (plural).
Because учитель is masculine singular, the verb takes the masculine past form заметил.

What case is учитель, and why isn’t there an ending on it?

учитель is in the nominative case (the subject). Many masculine nouns in the nominative singular have zero ending (no visible ending), so the form stays учитель.

Why is it ошибку and not ошибка?

ошибку is accusative singular, because it is the direct object of заметил (noticed what?).
For many feminine nouns ending in -а, the accusative singular ends in -у:
ошибка (Nom.) → ошибку (Acc.).

Why is it в моём докладе—what case is докладе?

After в meaning “in/inside,” Russian typically uses the prepositional case for location.
So доклад becomes докладе (Prepositional singular): в (где?) докладе.
The phrase means “in my report,” i.e., the error is located within the report.

Why does моём look different from мой?

мой changes to agree with the noun’s gender, number, and case.
Here it modifies докладе (masculine, singular, prepositional), so it becomes моём:

мой доклад (Nom.)
в моём докладе (Prep.)

Is моём the same as моем? What’s the deal with ё?

Yes in meaning: моём and моем are often the same word in print, because ё is frequently written as е.
But the pronunciation differs: моём is pronounced with yo. In careful writing (especially for learners), ё helps avoid ambiguity.

Why is it попросил меня исправить её—why is меня in that form?

меня is the accusative (and also genitive) form of я. With попросить (“to ask/request someone”), the person asked is typically in the accusative:
попросил (кого?) меня.

Why do we use an infinitive after попросил instead of a form like “that I correct…”?

Russian commonly uses the pattern попросить + (кого?) + infinitive:
попросил меня исправить = “asked me to correct.”
You can also say it with a clause, especially if you want to state the subject explicitly:
попросил, чтобы я исправил её (more like “asked that I correct it”).

Why is the infinitive исправить (perfective) and not исправлять (imperfective)?

исправить is perfective, emphasizing a completed result: “to fix/correct it (and finish correcting it).”
исправлять is imperfective, emphasizing the process or repeated action: “to be correcting / to correct (in general).”
In a one-time request to fix a specific error, perfective исправить is the natural choice.

What does её refer to, and why is it in that form?

её means her / it (accusative/genitive). Here it refers to ошибку (a feminine noun), so English “it” corresponds to Russian её (because the noun is grammatically feminine).
It’s the direct object of исправить: исправить (что?) её.

Could you drop её and just say исправить?

Often yes, if it’s obvious from context: …попросил меня исправить can work in conversation.
But including её makes the sentence clearer and more complete in writing, explicitly pointing back to ошибку.

Is the word order fixed here? Could it be rearranged?

Russian word order is flexible, and changes often shift focus/emphasis rather than basic meaning. For example:

Учитель заметил ошибку в моём докладе и попросил меня исправить её. (neutral)
В моём докладе учитель заметил ошибку… (emphasizes where)
Учитель попросил меня исправить её, заметив ошибку… (more literary; emphasizes the request)
The given order is very natural and neutral.

What’s the difference between заметил ошибку and something like нашёл ошибку?

Both can translate as “found/noticed an error,” but the nuance differs:

заметил ошибку = noticed it (it caught his attention)
нашёл ошибку = found it (more like discovered it, perhaps after looking/checking)
In a teacher context, заметил is common: he noticed an error while reading.