since Windows 2012 we know that it is not necessary to use a heartbeat NIC in a cluster as far as I can remember.
No actually it is from Windows Server 2008 that Heartbeat is not mandatory.
Do I need to use a NIC for heartbeat between nodes 1 and 2?
Not necessarily your WSFC will just work fine if you do not have Heartbeat NIC or heartbeat Configured. I answered a very similar question on Stackexchange before, have a look. But if your NIC which hosts both cluster and client communication gets saturated more often I would suggest you set up heartbeat. It does not adds complexity.
Is there a need for DR sites as well? If so, how would that be possible since they are different networks?
Again depends but configuring that is not a pain, you need to add a network adapter on DR and add a connection. Your network team can easily set it up for you.