Is connecting the ESXi host to the NAS directly doable to begin with?
Yes, should be. Still you can try to test it as I don't find any reference article with steps.
How will I be able to assign a static IP to the new added NIC on the ESXi host? Will it show up in the IPv4 settings?
You have to create a vmkernel portgroup in order to configure network on ESXi node.
I can aggregate two ports on the NAS - can I also do that on the ESXi side? If not, I guess there's no reason for two transceivers on the server other than a spare?
In the virtual switch settings you can set a teaming policy for nics and make them active/active.