После вебинара я поеду домой на метро.

Because после requires the noun to be in the genitive case (“after what?”).

• вебинар (nominative) → вебинара (genitive singular)
  So После вебинара = “After the webinar.”

Yes, вебинар is masculine. Common clues:

• It ends in a consonant (-р), which is typically masculine in Russian.
• In dictionaries it’s listed as вебинар, -а showing the genitive ending -а, typical for many masculine nouns.

Because поеду is the future form (perfective) meaning “I will go / I will ride (by transport).”

• я еду = “I’m going / I’m riding” (present, happening now)
• я поеду = “I will go (once, in the future)”
  Russian often chooses a perfective verb for a single future trip.

Yes. поеду is the 1st person singular future of поехать, which is the perfective partner of ехать.

• ехать (imperfective): to go/ride by vehicle (process)
• поехать (perfective): to set off / to go (as a completed, single trip)

The verb ending -у already shows “I,” so поеду can stand alone.
But я is often used for:

• emphasis/contrast (“I will go, not someone else”)
• clarity in longer sentences
  So both are possible: После вебинара поеду домой на метро is also natural.

домой is an adverb of direction meaning “(to) home.” It answers “where to?” (куда?) and implies movement toward home.
Compare:

• домой = “(go) home” (direction)
• дома = “at home” (location, no movement)

домой is the standard, idiomatic way to say “home” as a destination (like English “go home,” not “go to the home”).
You can say в дом but it usually means “into the house (building),” often implying entering a specific building, not just “home” as a concept.

Literally it’s like “by/on the metro,” meaning “using the metro as transport.”
Russian commonly uses на + (prepositional) to express the means of transport with many vehicles:

• на метро, на автобусе, на трамвае, на машине
  So на here is the typical preposition for “by (transport).”

метро is usually indeclinable (it doesn’t change form in cases).
So you say:

• в метро, на метро, из метро — still метро
  It’s a borrowed noun ending in -о that stays the same.

на метре would refer to метр (“meter”) in the prepositional case, meaning “on a meter” / “at the one-meter mark” depending on context.
It would not mean “by metro.” The transport word is метро.

Russian word order is flexible, but changes emphasis:

• После вебинара я поеду домой на метро. (neutral)
• Я после вебинара поеду домой на метро. (emphasizes “I” slightly)
• После вебинара я на метро поеду домой. (emphasizes “by metro”) Most versions are grammatically fine; the original is very natural.

It’s common and standard in modern Russian, especially in professional/online learning contexts. It’s a loanword, but widely used and acceptable in formal and neutral speech.

Yes, but it changes the meaning:

• пойду домой = “I will go home (on foot / as a general ‘go’)”
• поеду домой = “I will go home (by transport/vehicle)”
  Since the sentence includes на метро, поеду is the natural choice.