Not specifically. ECDSA is a signing algorithm. Unless there's some really fancy asymmetric authentication protocol between components within the phone, which perform curve operations, it's most unlikely. Of course, there's no saying a hardware side-channel isn't a possible attack vector.
There is a dedicated authentication coprocessor, to which the communications will be authenticated, so yes.
The thing you're talking about ("secure enclave") is not present in the phone the FBI are trying to attack.
Also, the Secure Enclave co-processor is on-die, so there will be no need to authenticate or encrypt communications with it.
I suppose not. I presume the way would be vulnerabilities in code handling untrusted data streams such as USB, Wi-Fi, or BlueTooth. Although the amount of time it takes might be getting longer, every major iOS update eventually has had jailbreaks, which implies for an organization with larger budget than jailbreak hackers would likely be able to discover 0-day vulnerabilities in iOS.