The UK is one of two countries in the world with unelected clerics as part of their national legislature, along with Iran.
Well, there's
always the Vatican.Though apparently they are lagging in this decision behind the likes of Canada, Australia and no other than America.
It's a bit complicated.
The Church of England is basically the originator and largest section of the Anglican denomination. However, different branches of the wider church have significant autonomy, in contrast to Catholicism. Right now Anglicanism in most of the English speaking world is a rather liberal denomination, which is reflected in their practices. However, once you get into the other major branches (notably the African churches) it turns considerably more conservative, especially on the Big Questions around women and homosexuality. The CofE has basically tried to split the difference to try to avoid too many churches splintering off into other denominations.
Within the Church of England itself you still have divisions, with the mainstream (liberals or broad church group) being pulled around by the low and high church factions. Low church (Evangelicals) groups traditionally put more emphasis on the Protestant tradition, and more recently include a lot of North American style conservatives and ideas. High church (Anglo-Catholic) groups put more emphasis on Catholic traditions. Needless to say, both of these factions can be resistant to liberalisation moves.
It's also important to note that the CofE is essentially the default religion in the UK, so it's hard to say what the views of its followers are. The committed, active members might hold very different views of the majority of those only vaguely aligned. Add into this its close entanglement with the (generally socially liberal) government and you have a very confused institution.