Deleting an asset removes it from all places, no traces remain. Even from rounds.
If you change properties of an asset, such as name, value etc. then most of these changes are seen everywhere, including in a division round. But currently other users will only see such changes when they refresh the page or their navigation causes the division data to be reloaded from the FairSplit server.
Not so for the Distribute column, however. When you create a round, its list of assets is created and frozen. Changing between Divide and other destinations only affects the next round that you create.
Adding an asset does not affect active rounds. This is on purpose -- it would be unfair if some divisees did not have time to bid on some assets.