What security settings should one use with Keychain

The system keychain is stored in /Library/Keychains/System.keychain and the key to unlock it is stored in /var/db/SystemKey (its default file permissions are readable by root only). The location of these files is referenced in the security-checksystem script (from the security_systemkeychain source). It is even possible to test to automatic locking/unlocking of the system keychain by using

systemkeychain -vt

The keychain security framework allows non-privileged programs to make requests for information provided they are in the ACL stored within the keychain entry. Obviously if a user has root they on a system they can directly access both the file storing the system keychain and the key to unlock it, thus they do not have make requests via the security framework and are not beholden to the ACLs stored within the keychain itself.

(I didn't actually answer the original questions so let's give this another go)

How are the keys architected such that any administrative user can unlock the System Keychain?

The libsecurity keychain framework allows regular processes to interact with the system keychain in an authenticated manner using Apple's XPC interprocess communication framework (IPC).

Program A sends a request to access the system keychain information using IPC. A check is made that the requesting user is already in the wheel group and also knows the password of a user in the wheel group. Once authorization is confirmed, the privileged kcproxy daemon can be used to access material in /var/db/SystemKey, unlock the system keychain and return the requested information.

Are there cryptographic restrictions that limit what an administrative user can do with information in the System Keychain in any way?

No - an administrative user is allowed to access/change anything in the system keychain. Even if they couldn't, they could copy the underlying files to another machine on which they have complete control and just unlock/access it there.

Given an unencrypted system backup without /Users, how would you gain access to the keys in the System Keychain?

If the backup contained copies of /Library/Keychains/System.keychain and /var/db/SystemKey then I would copy them to their respective locations on a new OS X system and use systemkeychain to make the later unlock the former and dump the keychain database using security dump-keychain.

The Apple TV (2nd and 3rd generations) runs iOS under the hood, so it almost certainly (hard to be 100% sure without doing some substantial hacking) uses the system keychain, just like iOS on an iPhone or iPad.

I defer to someone with a more concrete knowledge of the iOS security layers, but my understanding is that like most other things on iOS, keychain access is sandboxed, so an app can only access the data that belongs to it. There are further security mechanisms in the keychain, but I don't know enough to speak about them with confidence. Based on the answers to this StackOverflow question, I think that it is possible to gain access to keychain data if you have physical access to the Apple TV, but it's not a trivial task.