Is it now possible to book trains?

Yes. I had never been on a train so I used Concur on a return trip year before last and booked an Amtrak train. I switched from a regular passenger coach cabin train to one where I had my room/car. It was less than a flight still and was definitely an experience. I dined at the dining car which was great and something that is apparently going away.

Yes, you are able to book rail tickets.

Yes, you can book trains. From my experience, Via rail in Canada.

Yes You can, as long as the link for the Amtrak is available in SAP Concur. You can go to the app and manage your trips. Otherwise, I sometimes book the train manually from the website and add the trip infomation manually to the SAP concur.

Yes, I've used it to book Amtrak before.

Yes, you can to book trains. The offer in Europe is insufficient.

it's still very limited, especially in Europe where rail travel is most common

Yes. When I request travel it gives me Amtrak trains and airlines on the same screen, unless I exclude the trains.