Can I detach the primary (eth0) network interface when the instance is running or stopped?

A.
Yes, You can.

B.
No. You cannot

C.
Depends on the state of the interface at the time

Explanation:
Each instance in a VPC has a default elastic network interface (the primary network interface,
eth0) that is assigned a private IP address from the IP address range of your VPC. You
cannot detach a primary network interface from an instance.