Learn Hardcore Russian: После обсуждения все были довольны новым расписанием. - After the discussion everyone was satisfied with the new schedule.

The base (dictionary) form is обсуждение (nominative). In the genitive singular, neuter nouns ending in -ие usually change to -ия → обсуждение → обсуждения.

Could we say После обсуждение instead of После обсуждения?

No. That would be grammatically wrong.

Because после always requires the genitive, you must change the noun from nominative обсуждение to genitive обсуждения.

Saying После обсуждение is like saying “After discussion (nominative)” in English instead of “After the discussion” – it violates the case rule that goes with после.

What is the difference between после обсуждения and something like после того, как мы обсудили…?

Both describe something happening after the discussion, but the structure is different:

после обсуждения – literally after (the) discussion, using a noun.
после того, как мы обсудили новое расписание… – literally after we discussed the new schedule, using a clause with a verb.

После обсуждения все были довольны… is more compact and neutral.
После того, как мы обсудили… can sound a bit more explicit or slightly more formal, and it emphasizes the action (“after we had discussed”) rather than the event as a thing.

What exactly does все mean here, and why do we use были (plural) with it?

In this sentence, все means everyone / all (people), so it refers to several people.
Because the subject is logically plural, the verb быть is also plural: все были.

Compare:

Все были довольны – Everyone was satisfied.
Всё было хорошо – Everything was fine. (всё = everything, neuter singular; verb is singular было)

So: все (everyone, all of them) → plural verb были.
всё (everything) → singular neuter verb было.

Can we leave out были and just say После обсуждения все довольны новым расписанием?

You can, but the meaning changes:

После обсуждения все были довольны… – past: at that time after the discussion, everyone was satisfied.
После обсуждения все довольны… – sounds like a general statement or like you are describing a current, typical result (“After discussion, everyone is (usually) satisfied with the new schedule.”)

If you are talking about one specific past situation, you normally keep были to show that it’s in the past.

What kind of word is довольны, and why is it in this form?

Довольны is a short-form adjective (краткая форма прилагательного), plural.
The full form is довольный / довольная / довольное / довольные.

Short-form adjectives are very common in predicative position (after быть, стать, остаться, etc.) to describe a temporary state:

Все были довольны. – Everyone was satisfied.
Он доволен. – He is satisfied.
Она была довольна. – She was satisfied.

You normally would not say все были довольные here; that sounds odd or changes the nuance. In this kind of “subject + was + satisfied” construction, the short form довольны is standard.

Why is новым расписанием in the instrumental case?

The adjective доволен / довольна / довольны always governs the instrumental case:

доволен чем? – satisfied with what?
довольны новым расписанием – satisfied with the new schedule.

So both words go into the instrumental:

новый (nom.) → новым (instr.)
расписание (nom.) → расписанием (instr.)

Other examples:

Я доволен результатом. – I am satisfied with the result.
Мы довольны работой. – We are satisfied with the work.

Could we say довольны новое расписание instead of довольны новым расписанием?

No. That would be wrong, because доволен / довольна / довольны requires the instrumental case, not the nominative or accusative.

You must answer the question довольны чем? (“satisfied with what?”) using the instrumental:

довольны новым расписанием – correct
довольны новое расписание – incorrect (wrong case)

Can we change the word order, for example: Все были довольны новым расписанием после обсуждения?

Yes, Russian word order is flexible. These variants are all possible and grammatical:

После обсуждения все были довольны новым расписанием.
Все были довольны новым расписанием после обсуждения.
Все после обсуждения были довольны новым расписанием.

The usual, neutral version is the original one, with После обсуждения at the beginning.
Moving после обсуждения can slightly change the rhythm or emphasis, but the meaning stays the same in this case.

Is обсуждение here a noun or some kind of verb form?

Обсуждение here is a noun meaning discussion. It is formed from the verb обсуждать / обсудить (“to discuss”) by adding the suffix -ние / -ение:

обсуждать → обсуждение – discussion
двигать → движение – movement
решать → решение – decision

So После обсуждения literally means after the discussion (event), not “after discussing” as a verb form, even though in English we often translate it with a gerund: “After discussing it, …”

What is the base form of расписанием, and how is it declined?

The base (dictionary) form is расписание (neuter noun, nominative singular).
In your sentence it appears in the instrumental singular as расписанием, because of довольны чем?

Key forms (singular):

Nominative: расписание – the schedule
Genitive: расписания – of the schedule
Dative: расписанию – to the schedule
Accusative: расписание – (same as nominative)
Instrumental: расписанием – with/by the schedule
Prepositional: о расписании – about the schedule

In новым расписанием, both the adjective and the noun match in instrumental singular.