7 is indeed indefensible; it is impossible - even in principle - to prove a negative, such as to know that something does not exist.
That's a bit of internet folk-wisdom, I think. Unfortunately, it isn't exactly true. There isn't any principle in formal logic that says 'one can't prove a negative'. Logicians do it all the time, it's totally routine.
For example, I want to know whether there's a sock in my drawer. So I open my drawer and look. There are three objects in the drawer and I look at each of them in turn, and none of them is a sock. So I demonstrate, by
simple enumeration in this example, that there's no sock in my drawer.
General statements about all X's can be disproven too, if the statements in question can be shown to imply a contradiction. That's a
reductio-ad-absurdem proof and they are routine in logic and mathematics.
It can get complicated, but it's possible to show using more advanced
model theory, that sometimes it's a lot easier to prove a negative than to prove a positive. That depends on issues like the nature of the universe of discourse.
But 1 is quite defensible.
I'm going to disagree with that one too.
1] It could - in principle - be proven true tomorrow, if the clouds parted and a colossal bearded man with a staff appeared and carried us all up into the clouds. (The same cannot be said for 7. We atheists will never have irrefutable proof.)
I think of this as the "Independence Day problem'. (The reference is to that 1990's science fiction movie where 15 mile wide flying saucers appear over the world's major cities.)
The thing is, no matter how impressive the light-show, we still don't know whether whatever is responsible for it is a god. In other words, we don't know whether the responsible party is a suitable object for religious worship.
The underlying philosophical issue is that events that transcend our human experience occupy a logical space that we can call
the unknown. If any gods exist, then presumably they are out there in the unknown too. The problem is that if everything out there is unknown, just by definition, we aren't going to have any means of distinguishing the divine stuff from the non-divine stuff.
2] It is often argued that there is "evidence" of God's existence all around us. The fact that us atheists people do not consider it convincing evidence does not mean that evidence was not brought forth. God is one explanation for the universe and life on Earth, it's just not one that atheists can get behind.
I think that the universe is probably
consistent with many different theologies. That's to be expected if the theologies were dreamed up by beings that are resident in the universe.
So, 7 ("I know God does not exist") is a flawed stance in both principle and practice.
It depends on how strongly we construe the word 'know'. If it means 'no possibility of error', then I agree with you. But probably no real-life human statement could ever satisfy that strong condition. If it means something weaker, like 'I personally believe with a high level of subjective certainty', then #7 is probably fine.
1 ("I know God exists") is a valid stance in principle, though in practice atheists and theists disagree.
I think that similar problems exist with that one too. As I suggested up above, there's a problem with human beings knowing whether any of their experiences is truly an experience of a god, a being worthy of religious worship. Believers might possess a high level of subjective certainty, but they can't evade the possibility of error.