I would say no since it isn't in play. You cant put a stasis on something that isn't there.
My guess as well. But, going thematically, how does it re-appear? So, it's clearly not destroyed. I'm thinking it's more like a powered shield and you've initially disabled the power. Then suddenly (when rolling a 5), the power is reconnected. So it's like sitting over in the corner waiting to be plugged in. So my stasis simply covers the power connection?
I don't know. It shouldn't be relevant anyway, because by the time you're there, you've won. I guess unless you're doing the variant of blow up core before guns wins.