yes machine key is a factor in how asp.net encrypts secure cookies and is a factor when using encrypted passwords.
machinekey should be unique and secret to an installation and should not be changed if using encrypted passwords
so could it be related, yes, do I know if it is related no. machinekey exists in web.config and would not normally be changed just by moving the site files to a new server, so one would not expect that to be the problem
definitely it is a configuration issue of some kind, but I have no definitive idea what it is. clearly something is different in the new configuration since it worked before and the files have not changed.
ssl configuration seems a likely possible cause so I mentioned it in the other post, but I have no way of knowing.
I would also try using different web browsers and different client machines, things like privacy/antivirus software or other security tools on a client machine can also prevent cookies