I thought wine was not to be run as root, is this not correct?
It's not the ICMP you need to look at but "SOCK_RAW". This is not supported for regular users, only superuser.
So to be clear: we strongly recommend against running Wine as root. But you sometimes need to run Wine as root in order to do certain things on Linux, e.g. open raw sockets. The solution: don't run applications that want to do those things, or take it up with the Linux kernel developers.
--Juan