Just applied another update. This time OW crashed after clicking "update and restart." It relaunched anyway, this time creating a new dock icon linked to the OmniWeb 5.6.app and the previous one was targeting the OW app in the trash.

Lemme give renaming the app a go and see how this works for me. That could be part of the problem here.

I have seen this too. I just made the update from 92419 to 92474 and it started the 92474 as is should, but if I command+click on the Dock icon to reveal the app in the Finder then it shows me "OmniWeb 612.0.92419.app" in the trash.

Part of this is just due to how launch services works. You should check for updates again and see if you're actually running the newest version or not.

I changed the name back to "OmniWeb.app" relaunched and when the next update came out, I ran the updater. The updater crashed. OW relaunched and again spawned a new OW icon and the original one now points to the OW install in the trash.

FYI: problem still persists here too.

I quit OW, deleted my history file and relaunched. When the final release updated last night, it updated smoothly.