Motherships can only spawn as part of an event-attack (either the FS chase groups, FS/Golem/Spirecraft exo-galactic strikeforce, or a FS AI-homeworld-defense spawn), and generally their point cost is so high that only the FS AI-homeworld-defense spawns have a chance of having a budget high enough and concentrated enough and lead-ship-heavy enough to afford one.
But if they do spawn, they will always have the core-shield-network-node effect. If you're in the final part of FS that automatically destroys the core-shield-network then the mothership will be insta-killed in less than 2 seconds (generally). I'm planning to put in a non core-shield variant for that case.
Anyway, I'm not sure what you mean by not being able to shoot at one. If you mean the no-autotargeting rule and thecan't-shoot-due-to-no-human-command-station rule, those rules don't apply to the mothership.