How to open the door to Alftand Cathedral

It is the Dark Brotherhood's sanctuary.

To get through that door you have to

kill the owner of the orphanage in Riften and then a courier brings you a note. Sleep to be contacted by them, then they will give you the necessary information to join from there.

They may not contact you the first time you sleep so keep trying, eventually they will.

When facing the gate they want you to open, turn to your right. There is a side tunnel. Follow that tunnel up some stairs and follow the hallway. You will see a pedestal to your right (which leads to the secret room you described, I believe). Walk past it to the end of the hall and turn a corner, and you will see a bridge which crosses over the lower room. At the end of the bridge you will find a rectangular room with a couple of shelves and a chest at the end. Ignore them for now, and look to your left. There will be some candles near a pull switch on the wall. Use that, and you'll be ready to continue.