Is Shiva a demigod?
Shiva is in a unique position. He's not an ordinary living entity (jiva), he's not an ordinary demigod (deva), nor is he in the Vishnu (God) category. Shiva has his own tattva, category of being, known as shiva-tattva. His residence, Kalilash, isn't annihilated when the rest of the material manifestation is annihilated.
Brahma, in his prayer known as Brahma-samhita, compares Shiva to yogurt and Krishna to milk. By adding some culture to milk, one makes yogurt. Both come from the same substance, yet are slightly different in quality. Milk, or Vishnu, is the source. Such is the relationship between Shiva and Krishna.
Lord Krishna (Vishnu) is responsible for the maintenance of the material and spiritual worlds. Lord Shiva is responsible for the destruction of the material world. He is also in charge of the tamas (ignorance) mode of nature, although he's not personally affected by it. Lord Shiva is constantly in meditation on Lord Vishnu; he's extremely powerful and full of good qualities. He's easily pleased by service and devotion to him, but he's most pleased when his followers direct their ultimate worship to Lord Vishnu.
Read the story of Daksha in the fourth Canto of the Srimad Bhagavatam; there you'll get a good understanding of the power and majesty of Lord Shiva.