The lack of teaming capabilities for the Ethernet ports is probably caused by two different reasons.
For Intel(R) I350 Gigabit Network Connection #2, this port probably has system management capabilities. Teaming is not supported on a port that can be used for server management, because the management engine takes precedence.IF you have a BIOS option to disable the management engine and if you don't plan to use that capability, you could try disabling it and then checking the capabilities.
I am not sure why the Intel(R) PRO/1000 PT Dual Port Server Adapter ports are not advertising the teaming capabilities. Have you tried creating a team with them even though the capability is not advertised? If you cannot get the teaming to work on these ports, then I would try uninstalling the Intel Network Connections software and then install the software again. I can not think of any good reason why you should not be able to team those ports.
I hope this helps.
I ended up rebuilding the operating system and without making any changes to the Bios all the cards are now showing the expected options. My Experience with Server Core over the last 4 years is that any hardware change typically requires an OS rebuild. This has been true on HP, Asus and Fujitsu server hardware. I think this is evident with the lack of tools available from Microsoft for maintaining software drivers in Server core installations.
The take away is that with server core installations is that they are purpose built appliances. Once they are deployed and used, any hardware changes should be accompanied with an OS re image.
In this case I had enabled the clustering feature and had used it in my Hyper-V cluster prior to adding the Dual Port cards.