If you decide to go for two different ports, then you don't need two network cards, you don't need two public IP addresses, and you don't need two different hostnames.
So 2 different ports, 1 network card, 1 public address.
In this case, i have to register for 1 domain ? (www.mycompany.com)
So what to do with the other ? (push.mycompany.com)