Learn Hardcore Russian: Студенты должны были сдать отчёт в январе, но преподаватель дал им ещё неделю. - The students had to hand in the report in January, but the teacher gave them another week.

So Студенты должны были сдать отчёт в январе ≈ The students were supposed to submit the report in January / had to submit the report in January.

Why is the verb сдать used instead of сдавать?

This is an aspect question.

сдать = perfective aspect
- focuses on a single, completed result: to hand in, to submit (once)
сдавать = imperfective aspect
- focuses on process or repeated actions: to be handing in, to submit regularly, to keep submitting

Here we are talking about one specific deadline and one completed action (the act of turning in the report). So Russian uses the perfective:

сдать отчёт = to submit the report (as one complete act)

If you said должны были сдавать отчёт, it would sound like they were supposed to be regularly submitting reports, or be in the process of submitting, which is not the idea here.

What case is отчёт in, and why does it look the same as in the dictionary form?

Отчёт is the direct object of сдать, so it is in the accusative case singular.

The dictionary form отчёт is nominative singular, but for masculine inanimate nouns, the accusative singular looks the same as the nominative:

nominative: отчёт (subject)
accusative: отчёт (direct object)

So there is actually a case change here (to accusative), it just happens to be identical in form.

Why is it в январе and not something like в январь?

After в, the case depends on meaning:

в
- accusative = movement into something
  - в январь would sound like into January (only in very special figurative contexts)
в
- prepositional = location in space or time
  - в январе = in January

Here we are talking about time (when the report had to be submitted), not movement, so we use prepositional case:

январь (nominative) → в январе (prepositional)

Why is it дал им and not дал их?

This is about cases and roles:

дать (to give) takes:
- dative for the person you give something to (indirect object)
- accusative for the thing being given (direct object)

In the sentence:

им = to them → dative plural of они
неделю = a week → accusative singular of неделя

So:

преподаватель дал им ещё неделю
= The teacher gave them another week.

If you said дал их, that would mean gave them (as things) to someone, which is wrong here.

What exactly does ещё неделю mean? Why is it not ещё неделя?

Ещё here means an additional or another.

неделя is feminine; accusative singular is неделю.
After дать, the direct object (what is given) is in accusative.

So:

ещё неделю = (for) another week / one more week

Ещё неделя in nominative would be used as a subject, like:

Ещё неделя, и экзамены начнутся.
One more week, and the exams will start.

In our sentence it’s an object of дал, so accusative неделю is required.

Could we say ещё одну неделю instead of ещё неделю? Is there any difference?

Yes, ещё одну неделю is also correct.

ещё неделю = another week, one more week (the one is implied)
ещё одну неделю = literally one more week, with одну making the one explicit

Nuance:

ещё неделю is a bit more neutral and common.
ещё одну неделю can sound slightly more emphatic or precise: exactly one more week (and no more).

Why is the conjunction но used here, and not и or а?

Но expresses contrast, often something unexpected or opposite.

но = but / however (stronger contrast)
и = and (just adds information)
а = but / whereas (lighter contrast, often between two states or subjects)

In ... в январе, но преподаватель дал им ещё неделю, the idea is:

They had to submit in January (expected situation),
but the teacher gave them more time (contrasting with the expectation).

So но is the natural choice.

И would sound like he gave them another week in addition, without the sense of “contrary to the plan”.
А is possible, but would feel weaker and more like just comparing situations, not clearly expressing the “instead / contrary to expectation” idea.

What is the difference between преподаватель, учитель, and профессор? Why use преподаватель here?

All three can translate as teacher, but they are used in different contexts:

учитель
- usually a school teacher (elementary/middle/high school)
- can also mean teacher in a very general sense
преподаватель
- someone who teaches in higher education (university, college, institute)
- can also mean instructor (e.g. at a language course, sometimes)
профессор
- a specific academic title: professor
- not just any teacher

In a university context (students submitting a report), преподаватель is the default, neutral word for teacher / instructor. Using учитель would suggest a school; профессор would be too specific.

Is the word order преподаватель дал им ещё неделю fixed, or can we move the words around?

Russian word order is fairly flexible. The neutral order here is:

преподаватель дал им ещё неделю
(subject – verb – indirect object – direct object)

Other orders are possible, depending on what you want to emphasize:

Преподаватель им дал ещё неделю.
Slight focus on им (to them, not to others).
Преподаватель дал ещё неделю им.
Also possible, with a stronger emphasis on им.

More unusual orders like:

Ещё неделю дал им преподаватель.

are also grammatically possible, but sound more stylistic/poetic or strongly emphasized.

For a normal, neutral sentence, преподаватель дал им ещё неделю is best.

How would the sentence change if we were talking about one female student instead of several students?

You need to change:

студенты (they, plural) → студентка (female student, singular)
должны были (plural) → должна была (feminine singular)

The rest stays the same:

Студентка должна была сдать отчёт в январе, но преподаватель дал ей ещё неделю.

Note also ей (to her) instead of им (to them), because:

ей = dative singular (to her)
им = dative plural (to them)