She is also very smart - has a masters in math, probably could easily answer trivia like how much of the Earth is covered in water. But she is firm in her beliefs and faith is always > reason.
This is not an attempt to insult your friend, but being good at math or some other subject does not necessarily mean you are a smart person. Being smart means you are capable of thinking critically and rationally about any subject, even ones you may not fully understand. But you will weigh the evidence objectively to form your opinion. I would personally not consider someone who believes in creationist garbage science, or someone who firmly believes that faith trumps reason a smart person.
That's a total non sequitor. Just because you can't prove the non-existence of faeries doesn't mean you get to assume they don't exist until someone does. There's no logic there at all.
You're going about this all wrong. The negative claim (fairies don't exist) cannot exist without the positive claim (fairies do exist) being made first. You can't say that you don't believe in god unless someone first makes the claim that god does exist. The claim and burden of proof both fall on the "god exists" camp.
Go back to your first statement and stick to it. The one making the claim has the burden of the proof. Any claim.
The claim is that god exists, not that he doesn't exist. Person A says to Person B, I believe god exists. Here's why. Person B says, your evidence is not sufficient enough to support your claim. Person B is not making a claim. He is rejecting the claim made by Person A based on lack of evidence.
The burden of proof is always on the one making the claim. You can't shift the burden of proof to the negative argument, because the negative cannot be proven. It's impossible to disprove the existence of fairies. It is possible to prove they exist, assuming you can find a live or dead fairy carcass.
In the case of god, what we have is a claim being made by believers that their god exists. They have to prove their case. Atheist or agnostics don't have to disprove it.
You are correct that the agnostic position should be the default one. But after you examine the evidence for a claim, you can judge the probability of that proposition being true. If the probability seems low enough, you can safely say that the claim is most likely not true given the current evidence.
We don't call it a religious belief if you deny the existence of fairies or unicorns, so why is it any different with god?
The burden of proof is on the believer. An atheist doesn't have to prove there is no God, the believer has to prove that there is. I've met no atheists that have said with 100% certainty that there is no god, simply because it cannot be disproved. Atheists just think there is not enough evidence to support the proposition, and the probability of there being such a being is very low.
I'm sure there are atheists who are 100% sure that god does not exist. But I've never met one. In fact, I bet if you took a poll among atheists, you would find that very few state that there is no god as a matter of fact.
One good reason why computers can do more work than people is that they never have to stop and answer the phone.