A family of five – two men, two women and a young child – were in the Glen Road property when two shots were fired at about 7.30am on Saturday.
Police said two men, believed to be involved, later attempted to hijack a car before making off on foot.
A 35-year-old man later suffered head injuries after being assaulted by three men in the same area.
His injuries are not believed to be life-threatening.
The men, both in their 30s, are helping police with their inquiries.
Detectives have appealed for anyone with information about the attacks to contact Dungannon Police Station on 0845 600 8000, or call Crimestoppers anonymously on 0800 555 111.