После вебинара я сел на поезд и поехал домой.

Because после requires the genitive case.

• вебинар = nominative singular
• после вебинара = after the webinar

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после встречи = after the meeting

So here вебинар changes to вебинара because it comes after после.

In the past tense, Russian verbs agree with the subject in gender and number.

Since the subject is я, the form depends on whether the speaker is male or female:

• male speaker: я сел, я поехал
• female speaker: я села, я поехала

So this sentence sounds like it was said by a male speaker. If a woman said it, it would be:

После вебинара я села на поезд и поехала домой.

Yes, сесть / сел literally means to sit down, but with forms of transport it often means to get on / board.

So:

• сесть на поезд = to get on the train / board the train
• сесть на автобус = to get on the bus
• сесть на самолёт = to board the plane

In English, we usually would not say I sat on the train in this context, but in Russian this is a normal way to express boarding transport.

With many kinds of transport, Russian idiomatically uses сесть на + accusative:

• сесть на поезд
• сесть на автобус
• сесть на трамвай
• сесть на самолёт

So на поезд is the standard expression for board a train.

A useful contrast is:

• сесть на поезд = get on / board the train
• ехать на поезде = travel by train

Notice how the case changes too:

• на поезд = accusative, direction onto it
• на поезде = prepositional, means by train

Because Russian often describes the sequence of actions more explicitly than English.

This sentence breaks the event into two parts:

1. я сел на поезд = I boarded the train
2. и поехал домой = and went/set off home

In English, we might compress this into something like After the webinar, I took the train home, but Russian naturally uses two actions here.

It gives a clear sense of order:

• first he got on the train,
• then he went home.

This is about aspect and also about verbs of motion.

• ехать / ехал is imperfective: was going, went (in process), was riding
• поехать / поехал is perfective: set off, went, started going

Here the speaker is talking about a completed, one-time action in a sequence of events, so поехал is the natural choice.

Compare:

• Я ехал домой два часа. = I was going home for two hours.
  (focus on the process/duration)
• Я поехал домой после вебинара. = I went / set off home after the webinar.
  (focus on the fact that the trip began and happened)

So поехал fits the idea of then I headed home.

Because домой is a special adverb meaning home / to home / homeward.

It is extremely common and does not need a preposition.

• идти домой = go home
• ехать домой = go home
• вернуться домой = return home

Compare:

• дома = at home
• домой = home, to home

So:

• Я дома. = I am at home.
• Я еду домой. = I am going home.

Sometimes, but the meaning would shift a bit.

• поехал = went, set off, started going
• уехал = left, went away, departed

In this sentence, поехал домой simply describes the movement home.
If you used уехал домой, the emphasis would be more on leaving the place after the webinar.

So:

• поехал домой = went home
• уехал домой = left for home / went off home

Both can be possible in some contexts, but поехал домой is more neutral here.

Yes. Вебинар is a borrowed word, but it behaves like a regular masculine noun in Russian.

Its singular forms are:

• nominative: вебинар
• genitive: вебинара
• dative: вебинару
• accusative: вебинар
• instrumental: вебинаром
• prepositional: о вебинаре

So после вебинара follows a normal declension pattern.

Because после вебинара is just a normal prepositional phrase at the beginning of the sentence, and Russian usually does not separate it with a comma.

So the standard punctuation is:

После вебинара я сел на поезд и поехал домой.

A comma would usually only appear if there were some special parenthetical structure or unusual intonation, which is not the case here.

Yes, Russian word order is flexible, though the original version is very natural.

Original: После вебинара я сел на поезд и поехал домой.

Possible variations:

• Я после вебинара сел на поезд и поехал домой.
• Я сел на поезд и поехал домой после вебинара.

These are all understandable, but they may shift emphasis slightly.

The original puts После вебинара first, which clearly sets the time frame right away: After the webinar...

Yes, that is also correct, but it means something slightly different in structure.

• я сел на поезд и поехал домой = I got on the train and went home
• я поехал домой на поезде = I went home by train

The second version focuses on the means of transport.
The original sentence focuses more on the sequence of actions: first boarding, then traveling.

So both are good Russian, but they are not exactly the same in nuance.