I assume there is no donor check on whether or not you are eligible to a double death chest, within the death chest code(apart from the inventory check). So my guess is that the double death chest is only applicable when you are holding chests in your inventory... unless that's a separate...
I thought for me double chests only spawn if you have some chests in your inventory. Because I seem to only get a single chest spawn without any in my inventory. Though I haven't thoroughly tested this.
Ancient runes will not stay in your inventory if you die and have a death chest spawn, it will go to the chest instead. If you die and your death chest can't spawn, it will stay with you, like it's meant to.