Well, if the tsar mentioned it was caused by coughing, that could well be. He could have got a really strong cold, and coughed so much he strained his inside which caused the bleed. (OTOH, the Tsar mightn't have written the real reason). Tatiana Botkin wasn't in the house after all, so she mightn't be fully accurate. If he did sled down the stairs (and though I think it is a myth, it is possible), it would be from boredom. Teenage boys do stupid things when they're bored. Even intelligent ones.