The word 'lingam' in sanskrit simply means just 'symbol'. Nothing to do with any private part. Some westerners have made up this story that Sivalingam is Siva's private part!
(Please read the book 'The art of God symbolisation' by Swami Chinmayananda).
Siva is the Lord of Yama himself and the 'Annihilator' among the Trinity of Hinduism. He always live in cremation grounds, covering himself with ashes (that is why Hindus wear the 'bhasma' or ashes on their foreheads). A good example is 'Kashi' or Varanasi which is nothing but a huge cremation ground!