RE: Taking a train on Shabbos.
Assume the “good old days” when you paid when you got on and could discard the ticket (or didn’t have one if a farebox was used and you paid with a coin or token), and assuming the train is operated by goyim, it would be possible to board a train before Shabbos (boarding after Shabbos would at the least require a Shabbos goy to pay your fare), and get off assuming you were on a train that made all stops so it would be opening the doors at your stop even in no one was getting on or off, and you were already Shabbos-dik, meaning not carrying anything. Under these conditions using the train to get home on Shabbos would probably be an option and I believe it was used fairly often in the “bad old days” before legislation 50 years ago introduced “reasonable accomodation” into American law. Today in that situation one has a problem since transit systems often require you to have a card to exit, and you would never be able to carry a fare card since it is clearly a cash equivalent.
An car or bus would not be similar since it would require adjust for number of passengers (knowing who connected a seat belt, avoiding smashing people when the door closes) and once the car is reacting to your presence it becomes an issue (unless one holds that elecronic automated systems are inherently legal on Shabbos, which is problamatic to say the least).