Learn Hardcore Russian: Мы обсуждаем наш выпускной день.

Imperfective: обсуждать → мы обсуждаем
Focus: process, duration, repeated action
Example idea: We are (in the process of) discussing our graduation day.
Perfective: обсудить → мы обсудим
Focus: result, completion
Example idea: We will (successfully) discuss / have discussed our graduation day.

So if you say:

Мы обсуждаем наш выпускной день. – we are currently discussing it.
Мы обсудим наш выпускной день. – we will (at some point) discuss it and finish doing so.

Why is there no preposition like о (about) before наш выпускной день?

With обсуждать, Russian uses a direct object in the accusative with no preposition.

обсуждать что? – to discuss what?
обсуждать план, идею, проект, день (to discuss a plan, an idea, a project, a day)

So:

Мы обсуждаем наш выпускной день. – literally We discuss our graduation day.

If you use говорить instead, you need a preposition:

Мы говорим о нашем выпускном дне. – We are talking about our graduation day.

So:

обсуждать что (no preposition)
говорить о чём (with о)

What case is наш выпускной день in, and why does it look like the nominative?

Наш выпускной день is in the accusative case, because it is the direct object of the verb обсуждаем.

Verb: обсуждать кого? что? → accusative
Direct object: наш выпускной день

However, день is masculine and inanimate, and for such nouns:

Nominative = Accusative in form.

So:

Nominative: наш выпускной день
Accusative: наш выпускной день

They look the same, but the function in the sentence is accusative (object).

Why is it наш and not наша or наше?

Наш agrees with день in gender, number, and case.

день is masculine singular.
In the accusative, with a masculine inanimate noun, the form is the same as nominative: наш.

So:

Masculine: наш день
Feminine: наша неделя (our week)
Neuter: наше утро (our morning)
Plural: наши дни (our days)

Here we have:

наш (masculine singular)
выпускной (masculine singular adjective)
день (masculine singular noun)

All three agree.

What exactly does выпускной день mean? Is it the same as prom or graduation?

Выпускной день literally means graduation day.

Important nuances:

выпускной день – the day of graduation in general.
выпускной вечер – the graduation party / prom night (very common phrase).
Often people just say выпускной and mean the graduation party / prom.

So:

наш выпускной день – our graduation day (the whole day of finishing school / university).
наш выпускной вечер – our graduation party / prom in the evening.

Could we also say Мы говорим о нашем выпускном дне? What is the difference?

Yes, you can say both, but they are not identical:

Мы обсуждаем наш выпускной день.
– We are discussing our graduation day
(more focused, purposeful, often more detailed or planned discussion).
Мы говорим о нашем выпускном дне.
– We are talking about our graduation day
(more general chatting; not necessarily focused or organized).

So обсуждать suggests:

analyzing
planning
going through details

Говорить о suggests:

just talking, chatting, mentioning.

Is обсуждать formal, or can it be used in casual conversation too?

Обсуждать is neutral in style. It is fine in:

casual conversation
Мы обсуждаем фильм. – We are discussing the movie.
more formal or professional contexts
Мы обсуждаем проект. – We are discussing the project.

For a more casual, less serious tone, people might also say:

Мы болтаем о нашем выпускном дне. – We are chatting about our graduation day.
Мы разговариваем о нашем выпускном дне. – We are talking about our graduation day.

Why is there no word like the in наш выпускной день?

Russian has no articles (no a / an / the).

In this sentence, наш (our) already makes the phrase specific:

наш выпускной день – our graduation day
(this already has a definite sense, similar to the in English).

If you removed наш:

Мы обсуждаем выпускной день. – We are discussing graduation day.
That could be more general, but still there is no article in Russian.

Can the word order change, for example Наш выпускной день мы обсуждаем? Would that be wrong?

You can change the word order in Russian; it will still be grammatical, but the focus changes.

Standard, neutral:

Мы обсуждаем наш выпускной день.
(focus on the action: we are discussing our graduation day)

Possible with emphasis:

Наш выпускной день мы обсуждаем.
Emphasis: our graduation day is what we are discussing (and not something else).

Russian allows flexible word order mainly for emphasis and information structure, not for grammatical reasons. The sentence in your example is neutral and natural as given.

How do you pronounce Мы обсуждаем наш выпускной день? Where is the stress?

Stress and rough transcription:

Мы обсуждáем наш выпускнóй день.
- мы – [mɨ]
- обсуждáем – [əpsʊʐˈdajɪm] (stress on -да-)
- наш – [naʂ]
- выпускнóй – [vɨpʊsˈknoj] (stress on -ной)
- день – [dʲenʲ]

Main stress points:

обсуддáем
выпускнóй

If we wanted to say our graduation days (plural), how would the phrase change?

You would make день plural and adjust everything to plural:

Singular: наш выпускной день – our graduation day
Plural: наши выпускные дни – our graduation days

Changes:

наш → наши (plural)
выпускной → выпускные (plural adjective)
день → дни (plural noun)

Then the full sentence in plural:

Мы обсуждаем наши выпускные дни. – We are discussing our graduation days.