1. Not really with concerns to statements to the non-existence of something. I'm sure many times in your life you have made such statements: "santa claus does not exist" without ever being under the impression that there is some onus on you to 'prove' that he doesn't. It only works the other way around - a positive claim to the existence of something. For example: "santa claus does exist". Interestingly, this applies to any unobserved, claimed existing entity.
2. It comes down to statement types. Consider the following:
- all swans are white
- there is a black swan in London
As you can see, the first statement is falsifiable, (show one non-white swan and it's falsified) but it isn't verifiable, (regardless to how much you search you could have missed a non-white swan).
The second statement is the opposite - it is verifiable, (you see the black swan) but not falsifiable, (if you don't see it that's because it flew somewhere else etc).
'god/leprechaun/fairy exists' statements are of the second type - they are verifiable, (show a god, leprechaun etc), but are not falsifiable, (if you don't see one it's because it's invisible etc) - hence the onus can only be upon the claimant to the existence of.
Now, along with fairies, dragons, demons, sasquatch and el chupacbra I can quite happily state that a god "doesn't exist" and go about my merry way until such time when you have evidence to show that this entity does exist. There is simply no reason to take the claims of theists as credible. If they contend otherwise I give them ample opportunity to make their case. Alas it typically ends up with, "it says in this book.." which is utterly pathetic.
nobody else can prove god to you with words or telling you experiences, you would need to prove it to yourself. howcomes you don't seem content in your words when you post about god.
peace.