Open the chest and then you will be able to drag items into the chest, in which they will then be removed from your inventory.
Hold shift and then right-click with the lever to place it without interacting with the chest.
Sometimes it is lag. But in this case, it seems like you have a block on top of your chest. If you put a block on your chest, the chest wont open.
After taming the horse and adding a saddle to it, you can add a chest by left or right clicking the horse, I forget which. Upon death, the horse should drop the chest and the contents of the chest.
You cannot place a chest on a horse.You can however, place one in the inventory of a Donkey.Simply sit on the tamed donkey and press E.This will open the donkey's inventory.Place your chest in the box that has the silhouette of a chest.
Simply place it down and right click on it. There, you will be able to put stuff in it and take stuff out.