I remember reading recently (from what I believe was a well-informed source, even though the specifics escape me) that the highest risk of injury in the event of a train accident is now from hot drinks causing scalds/burns when they go flying. It's not the biggest source of injury on the railway - I believe that goes to the ever-present category of slips, trips and falls.
However, by extension, I would expect that appropriate mitigation would be put in place for dealing with any potential accidents involving hot drinks, perhaps due to rough rides or the general movement of trains at any speed. I must admit I can't recall ever personally having spilt a hot drink on anybody, and I do a huge annual rail mileage in the company of tea and coffee, but I've come very close to it, and I'm used to balancing on trains - and I have seen it done by other people a number of times, with the minimum level of inconvenience usually being ruined clothes. In addition, I do remember a number of occasions when I've got back to a cab or locker on a train and found that my securely-stowed cup of coffee has worked its way free, and gone and spilt all over the place. Sometimes this has included dedicated cup-holders failing to live up to their jobs.
A paper bag is usually recyclable, unless heavily contaminated, in which case it should biodegrade fairly readily, and is probably less harmful than the waste generated by the bin bag you put the coffee cup in. It's also a surprisingly good way of carrying coffee - although they don't have outlets on trains, some shops like Pret have really got the skill of packing them into paper bags down to a fine art.