Learn Hardcore Russian: Окно было открыто, но в комнате всё равно душно. - The window was open, but the room is still stuffy.

Открыто here is a short-form passive participle (краткое страдательное причастие) used as a predicate: (the window) was open.
Открытое would be the long-form participle/adjective (полная форма) and usually describes a noun inside a noun phrase, e.g. открытое окно = an open window (as a description, not the main predicate).

What exactly is было doing here? Can it be omitted?

Было is the past tense of быть (to be) and it marks past time for the state открыто.
In the present, Russian usually drops to be, so you’d commonly say: Окно открыто.
In the past, you typically keep it: Окно было открыто. In casual speech, it can be dropped sometimes if the past is clear from context, but that’s less neutral.

Why is открыто neuter? Is it agreeing with окно?

Yes. Окно is neuter singular, and short-form participles agree with the subject in gender/number:

окно открыто (neuter)
дверь открыта (feminine)
балкон открыт (masculine)
окна открыты (plural)

Is this an example of passive voice? Who opened the window?

Grammatically, было открыто is a passive-type predicate focusing on the resulting state (the window being in an open state), not on the doer. The sentence doesn’t say who opened it.
If you wanted to name an agent, Russian can add it (less common in everyday speech), e.g. Окно было открыто мной (by me).

How is душно working grammatically? There’s no verb or subject.

Душно is a predicative adverb / category of state (категория состояния). It often forms an impersonal sentence: it expresses a condition in general, not tied to a grammatical subject.
So в комнате душно is structurally like (it is) stuffy in the room, even though Russian doesn’t use a subject like it.

Why is it в комнате (Prepositional) and not something else?

Because this is location with в + Prepositional to mean in (a place):

в комнате = in the room
If it were motion into the room, you’d use в + Accusative: в комнату.

What does всё равно mean here, and where does it usually go in the sentence?

Всё равно means all the same / anyway / nonetheless and signals that the second clause holds despite the first clause.
It’s flexible in position, but commonly sits near what it modifies:

...но в комнате всё равно душно (very natural)
You could also say ...но всё равно в комнате душно with slightly more emphasis on anyway.

Why is there a comma before но?

Because this is a compound sentence with two independent clauses joined by но (but):
1) Окно было открыто
2) в комнате всё равно душно
Russian normally separates such clauses with a comma.

Could I replace но with а here? What changes?

Sometimes yes, but the nuance changes:

но usually expresses a stronger contrast / contradiction of expectation: the window was open, but it’s still stuffy.
а is often a softer contrast/transition (sometimes like whereas/and). In this context, но fits best because it highlights the “despite that” relationship.

Why is всё spelled with ё? What if I write все равно?

Correct spelling is всё равно (with ё) because it’s from всё = everything / all.
In many texts, ё is written as е, so you’ll often see все равно. It’s usually understood correctly from context, but learners benefit from remembering the underlying form is всё.

Could the first part also be said as окно было открытым? Is that the same?

Окно было открыто is the most standard, neutral way to say the window was open (state/result).
Окно было открытым uses the instrumental (открытым) and can sound more descriptive or contrastive, sometimes implying “it was in an open state (as opposed to some other state).” In many everyday contexts, открыто is preferred.

If I want to say “The window was open (because someone opened it),” should I use a different verb?

If you want to emphasize the event of opening rather than the resulting state, you’d typically use an active verb:

Окно открыли = someone opened the window (doer unspecified)
Я открыл(а) окно = I opened the window
Whereas Окно было открыто mainly presents the state (it stood open).