It sounds like you want to configure a cookie insert persistence profile (probably with no timeout to use a session cookie). The BIG-IP would insert a session cookie in the response to the client whenever the client doesn't include the persistence cookie in that request.
As for item 2, do both domains resolve to VIPs on the BIG-IP? If so, do both VIPs point to the same pool? If both of those are true, then it might be possible to persist the same client to the same server. If not, can you explain what type of persistence behavior you're trying to achieve between the two domains?
Aaron
Nimbostratus
