I'm not so much a great thinker but I'll try to have a discussion anyway. I enjoy talking about such things though of being as dumb as rock.
When nothing exists, nothingness exists.
I truly have no words to explain my thoughts on this. Nothing doesn't exist, there is simply nothing. The apple I'm holding right now doesn't exist, but nothingness doesn't exist either. We say something exists if it is, but if there is none, it doesn't exist either.

For example objects exist because we may proove their existence, we see them and we may feel them. But these objects since they cannot think nor have senses, they do not exist for themselves. They exist but at the same time they don't exist. Something doesn't exist in true sense if there is nothing to proove their existence. They don't exist nor nothingness doesn't exist also. For example you can distinguish between good and bad because you know which one is which one. You have something to campare and then draw a conclusion whether this act is 'good' or 'bad'. But what if this universe is the only one, and suppose it doesn't exist. Then there is simply nothing - no time - no matter - no space. Not a single atom of you exists so there is no person to proove existence. Nor there is anything to compare with therefore nothingness doesn't exist (since you don't know what it is) nor matter doesn't exist nor time and etc
Is there a state where existence doesn't exist?
I think such a state is when you have no brain, and senses. For yourself you do not exist but there is something to proove your passive existence outside. Like in case of us and a rock for example.
Is there a state where nothingness doesn't exist?
I think it is when there is also not a single thing that exists, like in above example if universe doesn't exist with all the time and matter, nothingness also doesn't exist. Nothingness is said upon whether it's here in this universe or it is not, but when the universe doesn't exist, nothingness also doesn't exist.
When I die, will nothingness continue to exist?
If you really have a soul which will keep all the senses of normal human, nothingness in sense of "is it here or not?" will continue to exist I think.
When I die, will existence continue to exist?
On the other hand if you don't have soul, existence will not exist for you anymore, just like in above example with rock. Existence will continue for others - living though.
Vaccum isn't nothingness. There's energy in timespace.
I also think that real vacuums aren't possible since you are enclosing nothingness in something. THerefore in this case nothingness exists. However as far as I predict, time will continue inside the vacuum, and some single particles of elecrons will still be in the vacuum. What does the energy come from?Sorry for being so dumb. First person to get a head ache of my response wins a chocolate.