Use VMXNET3. I have seen the exact same problem more times than I can count.
I know the e1000 family is listed as supported for 2012 but it never works. Well it usually works for a while then stops.
I have some another Windows 2012 VMs with E1000 NIC, but they never have this problem. So i just confused it is related with E1000 and window2012 or other issue.
Sorry I don't have the exact "why" or "where" for it.
I have just encountered it many times and thought I would mention how we resolve it each time.
Some more info.
Another reason not to use it if you are using the listed versions of ESXi.