1. How does God know there's nothing [he] doesn't know [he] doesn't know?
Are we assuming an existent God with particular attributes? Of course this is all speculation but I'll take a stab at it....
So, if we assume an existent God along the lines of a monotheistic one, this is a question on the same order of "How do WE KNOW we exist?" Wouldn't we presuppose a form of self evidence?
Self evidence is found in the theoretical construct of how some have defined the being which is God.
We can easily conceive of something as human beings that we do not or cannot know. In other words we are self aware of our own limitations. In the same vein of thinking God would be that being which is self aware of his own infinite capacities as concerns that which is possible. It would be epistemologically self evident to God that there is nothing that God does not know that is possible to know. I would say that as finite beings we haven't the capacity to understand the mechanisms by which an infinite being would know that it knows all that can be known which would necessarily preclude the possibility of something unknown to the knower.
2. How does God know [he] didn't spontaneously spring into existence. fully formed with memories and all, with the rest of the universe last Thursday?
This is mostly just a stream of consciousness reply. Something to thresh out.
To answer this I think we need a conception of what temporality consists of in relation to self aware beings, specifically God (being the only singular element within its particular set, and the set containing all other self aware elements of reality). If temporality consists of recognizable changes in relations between differentiable things but Gods self awareness includes awareness of nothing differentiable within itself except those things which change in relation to things other than God, then temporality in relation to existence and non-existence would be meaningless as applying to a conception of Gods particular status.
Incidentally, it is a tenet of Christianity which believes that all of temporal existence (past, present, future) happened in one divinely creative moment. Hence Gods atemporality. Time does not "pass" for God. Time is not an "experience" God has. It is an experience mankind has. Time unfolds within mans awareness of his own existence. However with God, reality presents itself to Gods infinite awareness atemporally. That is in its entirety (past, present, future) in one infinite moment.
Our awareness of our existence is in direct relationship with our awareness of our temporal condition and the finite conditions encompassing that awareness. Our presumptions of our previous and potentially future non-existence is in direct relation to observations of other temporal things in a temporal framework and our awareness of the finiteness of those observations. Our experience of reality is temporally derived. Because it is temporally derived it is finite. Because it is finite it is epistemologically equivalent to say everything, memories and all popped into temporal existence at once versus over a real extended temporally evolved frame of reference. That is, it is equivalent to temporal beings.
Gods awareness however, being atemporally derived is in direct relationship to Gods awareness of its singularly unchanging divine nature
and its infinite knowledge of that awareness as suggested in 1 above. Gods existence as derived atemporally would not have a "when it began" so it would be nonsensical to say God sprang into existence "fully formed with memories etc." versus God simply never not existed atemporally fully formed memories and all.
IF God is an atemporal thing then questions of existence or non-existence cannot be sensibly applied to questions about God. You couldn't say...God exists now but not then or then but not now. That would be absolutely meaningless outside of the existence of a temporal standard by which a temporal beings changing state could be used in comparison.
So if the answer to question 1 above is somewhat true then God would be aware of its absolutely unique atemporal state and the temporal states of temporally dependent beings in comparison.
3. How does God know [he]'s not just a dream in the brain of a human?
I would refer back to 1 to perhaps help answer this question.