Tony, there are 3 places luggage goes in trains.
1) In racks above seats (which don't have doors)
2) Between the back of two back-to-back seats. This is an A-shaped space and is better for larger items.
3) One some newer long distance trains only, extra floor to ceiling luggage racks at the end or in the middle of the coach
More info, and pictures of all 3 spaces here: https://www.seat61.com/luggage-on-european-trains.htm
Second class is 2+2 seating, First class is 1+2 seating with proportionally wider seats and slightly more legroom. There is no extra luggage space in first class, but fewer passengers per coach.
"Can a large suitcase be checked?" - Trains do not have checked luggage or a baggage car (except in Hollywood films, they never do their research). All luggage is hand luggage, with no weight limits, except you have to carry it yourself, so the limit is your muscles.
Sometimes (e.g. between manned stations within Switzerland) you can send luggage ahead, but it doesn't travel on the same trains as you.