In real life, yes. In the Whoniverse, the Torchwood Institute seemed to be an exception of sorts. But perhaps Prime Minister Harriet Jones would have wrestled control of Torchwood away from Yvonne Hartman and thus prevented the Battle of Canary Wharf in the first place.
Exactly. Torchwood were running themselves, a rogue organisation who could claim legitimacy from Queen Victoria. I doubt Lizzie was sitting in Sandringham interviewing candidates for head of Torchwood.