We would like the possibility to set one absolute edit/admin url (http/https) for all sites in an multi site environment.
This to let the network admins to just open one url for editing and for editors that uses multiple to sites not have to re-authenticate when browsing the tree in edit mode.
Could be done by applicationsettings or in admin mode site settings.
That is not possible, since each website must map against one host. What you can do instead, is to have them all on different subdomains, since cookies can be shared across subdomains.
We thought of that possibility but the network admins for the customer says that we cannot use subdomains this way since they only want to open edit mode on a single specific domain outside their network.
We couldn't use wildcard cookies either since some sites have their own domain names which are not subdomains.
Regarding your last paragraph... that shouldn't be a problem, since a website can map against multiple hosts. So you can still have a primary host, which public visitor uses and then a subdomain for editors.
Let's say you have these hosts mapped to three different websites in Episerver admin:
Then you can also map these hosts to the same websites in Episerver admin:
company1.company.com -> www.company1.com
company2.company.com -> www.company2.com
company3.company.com -> www.company3.com
Then you need to configure the authentication cookie to work for subdomain.
But since your client won't allow you to have more than one host open in the firewall, you're out of luck.