your Lily is Haemanthus albiflos from Southern Africa. It is often called Paintbrush because of the shape and appearance of the flower - or Elephant's ears because of the foliage. It has been awarded the RHS Award of Garden Merit. It flowers best when potbound and is a really useful houseplant as it thrives on neglect! The best way to propagate is by careful division of any offset plants. It can be raised from seeds if your plant has ever fruited?
Tidying up the dead leaves is sensible. It will prefer an open sandy loam soil - but something like a John Innes soil-based potting compost should be fine.