Are housing authorities government agencies?

Not really; Housing Authorities are independent, not-for-profit public corporations. However, Housing Authorities get their funding from Housing and Urban Development (HUD) which is a federal government agency (and also other local sources).

The Housing Authority board members are picked by local government, but the Housing Authority must follow the rules and procedures set for them by HUD. They also are audited by HUD, and complaints about a Housing Authority made to HUD will be investigated and send an answer and in some cases, a plan to remedy any problems, back to HUD.