Ok, to avoid misunderstandings, I'm agnostic but epsitemologically liberal 
1. Instead of proof, I'd like to use word "evidence". Proof suggests that it forces you to accept the conclusion. However there is no this kind of evidence. The closest you can get is mathematics. Natural sciences and such things work only on evidence, which suggests that some theory is approximately true. This is clear from two facts: i) natural sciences are inductive, and ii) natural sciences have proved out to been erroneous.
Generally we don't require proof in oder to believe in somehting, evidence is good enough, and often we don't require even that.
2. There's evidence for everyone, and evidence for singular person. If you want others to believe something, you must provide evidence they have access. However you don't have to be able to convince others to be justified in your belief on something.
For example I have a coffee cup in front of me right now. I can't give any evidence for this thing. And even if you came here to look for yourself, I could have put the cup there later to fool you. There is no way I can give any evidence for the coffee cup's location when I'm writing this, but yet I am justified to believe it's on my table.
I believe this is exactly what religion is about: religious people have some sort of personal evidence, good or bad, and they ahve every reason and right to believe because of it. They can't communicate that evidence to others, but it doesn't mean it were not valid for them.
I recently noticed that it's difficult for people to think that someone might have an opinion without thinking it should be others opinion too: for example if someone doesn't eat meat, people often tacitly assume that he thinks no one else should eat meat either.
Even if you wouldn't think this way, there might be some remains of that sort of thinking in your argument: that you assume that evidence for religion should be evidence for everyone. And also the tradition of religious conversation supports that kind of view: religious people who wanted others to believe have used only arguments that they think should persuade others, but not stated the reasons they themselves believe.
The point of this whole thing is:If someone on street requires you to believe in God, it's ridiculous, for the exact reasons you stated in the OP. But if someone by his own believes in God, same kind of reasoning doesn't apply, since he doesn't require you to believe. He may have other reasons to believe than faith alone.
3. The idea of first principle is simplistic. People don't form their views from first principles. They rather have views, on which they base other views, and sometimes they re-evaluate some of them. These views don't form simple chains or "trees", and they aren't well thought out.
In the case of religion, I suppose, people haven't first read the bible and then decided to believe in God, or first believed, and then read the bible. Instead the bible and their personal experiences support each others, little like when you're building a card hous, and place two cards leaning against each others: /\.
Tyipcal western christian, I suppose, has been exposed to the stories of bible fron his very childhood. He might have never read or thought it much. Maybe he's got somekind of feeling about it. Then he might have some sort of experience that pushes him towards the bible at age of 18, and read it a little bit, forget about it, and read it again when he's 23, and see the whole thing in a new light. Then he might forget it again for some years, and have some bigger religious experience that makes him certain of the thing.
This is how people form their opinions on just about everything: no part is fundamental, instead different parts rely on each others. My belief on the coffee cup on the table relies on the reliability of my sense perception, but it's the actual sense perception on which I base the assumption of that reliability.
I believe this applies foe science too, although I must admit that I can't (at least here) give any comprehensive evidence for that. Mainly, you can check for the definitions of different measurment units, and notice that many of them incorporate some kind of scientific theory. Or you can think about different experiments or equipment used to perform them, and how they rely on scientific theories.

1. Instead of proof, I'd like to use word "evidence". Proof suggests that it forces you to accept the conclusion. However there is no this kind of evidence. The closest you can get is mathematics. Natural sciences and such things work only on evidence, which suggests that some theory is approximately true. This is clear from two facts: i) natural sciences are inductive, and ii) natural sciences have proved out to been erroneous.
Generally we don't require proof in oder to believe in somehting, evidence is good enough, and often we don't require even that.
2. There's evidence for everyone, and evidence for singular person. If you want others to believe something, you must provide evidence they have access. However you don't have to be able to convince others to be justified in your belief on something.
For example I have a coffee cup in front of me right now. I can't give any evidence for this thing. And even if you came here to look for yourself, I could have put the cup there later to fool you. There is no way I can give any evidence for the coffee cup's location when I'm writing this, but yet I am justified to believe it's on my table.
I believe this is exactly what religion is about: religious people have some sort of personal evidence, good or bad, and they ahve every reason and right to believe because of it. They can't communicate that evidence to others, but it doesn't mean it were not valid for them.
I recently noticed that it's difficult for people to think that someone might have an opinion without thinking it should be others opinion too: for example if someone doesn't eat meat, people often tacitly assume that he thinks no one else should eat meat either.
Even if you wouldn't think this way, there might be some remains of that sort of thinking in your argument: that you assume that evidence for religion should be evidence for everyone. And also the tradition of religious conversation supports that kind of view: religious people who wanted others to believe have used only arguments that they think should persuade others, but not stated the reasons they themselves believe.
The point of this whole thing is:If someone on street requires you to believe in God, it's ridiculous, for the exact reasons you stated in the OP. But if someone by his own believes in God, same kind of reasoning doesn't apply, since he doesn't require you to believe. He may have other reasons to believe than faith alone.
3. The idea of first principle is simplistic. People don't form their views from first principles. They rather have views, on which they base other views, and sometimes they re-evaluate some of them. These views don't form simple chains or "trees", and they aren't well thought out.
In the case of religion, I suppose, people haven't first read the bible and then decided to believe in God, or first believed, and then read the bible. Instead the bible and their personal experiences support each others, little like when you're building a card hous, and place two cards leaning against each others: /\.
Tyipcal western christian, I suppose, has been exposed to the stories of bible fron his very childhood. He might have never read or thought it much. Maybe he's got somekind of feeling about it. Then he might have some sort of experience that pushes him towards the bible at age of 18, and read it a little bit, forget about it, and read it again when he's 23, and see the whole thing in a new light. Then he might forget it again for some years, and have some bigger religious experience that makes him certain of the thing.
This is how people form their opinions on just about everything: no part is fundamental, instead different parts rely on each others. My belief on the coffee cup on the table relies on the reliability of my sense perception, but it's the actual sense perception on which I base the assumption of that reliability.
I believe this applies foe science too, although I must admit that I can't (at least here) give any comprehensive evidence for that. Mainly, you can check for the definitions of different measurment units, and notice that many of them incorporate some kind of scientific theory. Or you can think about different experiments or equipment used to perform them, and how they rely on scientific theories.