Q: Can we count on faith in believing serious detailed claims?
First, it is not unreasonable for a Religion or other group to propose a large, detailed body of beliefs. The problems of life are many and complex, and any philosophy that purports to explain, much less solve them, will have to be complex. If there is a God that intends to help people, that God must communicate a large set of beliefs to guide his followers.
Second, not all religions are the same in regard to how they view the proper role of human reason in the life of their followers. For example, according to the Bible:
Come now, and let us reason together, saith the Lord: though your sins
be as scarlet, they shall be as white as snow; though they be red like
crimson, they shall be as wool. (Isaiah 1:18 KJV)
Given this and other statements, both Judaism and Christianity accord ample space for the exercise of reason in the life of the believer in grappling with God's words and trying to understand, defend and apply them. It is not faith, but faith plus reason, that guides our life. Another verse is:
O taste and see that the Lord is good: blessed is the man that trusteth in him. (Psalm 34:8 KJV)
The above quote suggest a third point. A person is not faulted for initially being skeptical of some of the religion's claims if they make a good faith effort to test them. The Bible teaches that if you follow its teachings, certain benefits will accrue, not as a law that is always applied woodenly (see the Book of Job, for example where a good man is in distress because he suffered grievous hardship despite his good conduct) but as a principle that more often than not will be proven out. Thus over time, you attempt to live out part of what the religion teaches, see benefits, then take on more of its requirements over time. Eventually, with sufficient evidence - not proof - such a person decides to trust all the rest on faith. This is like opening an egg carton at the store and picking up two eggs to see if they are cracked. If they aren't you assume all the others are good. Mathematically, such partial testing can be very accurate if you perform enough tests.
Fourth, we do not have direct access to the experiences of people who lived long ago. This means that there will always be statements based on historical events that cannot be verified. Archeology can rule out some as unsupported or contradicted and make others plausible, but that is all. Must God repeat His miracles for each new generation? In Deuteronomy, it says:
These commandments that I give you today are to be on your hearts. 7
Impress them on your children. Talk about them when you sit at home
and when you walk along the road, when you lie down and when you get
up. 8 Tie them as symbols on your hands and bind them on your
foreheads. 9 Write them on the doorframes of your houses and on your
gates. (Deuteronomy 6:6-9)
We do not expect our children to understand physics and chemistry when we tell them not to touch the fire. Likewise, there is the recognition that each generation must teach the next. This is true of our science as well as our religion.
Q: To what extent a claim can be extensive in order for us to be
justified in counting on faith in believing it?
Fifth, we can ask if the intrusiveness, size, or other measure of the importance of a claim is backed up by events that certify its authority. Not that we can prove that those authoritative events happened (because of point four) but can we at least say that IF those events happened, that would be enough to "prove" the legitimacy of the claim? The flood of Noah, the destruction of Sodom and Gomorrah, the parting of the Red Sea, the exile and return, the raising of the dead, prophecies of the future - these are all spectacular events, all connected with changes in covenants, new teachings etc. If they happened, would that be sufficient evidence to make belief reasonable to those who witnessed those events? If so, then each of the pieces of the religion as they were added had substantiation and authority. It is only the trustworthiness of generations of believers in that faith accurately passing down those teachings that is the subject of faith, not the original commands themselves. We know with certainty that some cultures lied when they wrote their histories, because we have the evidence. The question then becomes, have the people of any cultures or religions recorded and transmitted the truth?
Nevertheless, a large demonstration of God's power is sufficient to justify God's requiring us to listen to and follow his words.
Q: What common examples of those are present in ordinary daily life?
To conclude, I will share a personal experience. There is no way you can verify whether what I am saying is true; it is an experience that can serve as a valid data point to me - a fraction of a "proof" - but will not be useable as part of any logical defense of Christianity to be shared with other people. Nevertheless, it points to an approach to religion that can become a proof for the individual who practices it.
About fifteen years ago, I lost all hearing in one ear. (Talk to my ear doctor and you could verify this claim.) After a month of treatment and several trips to the specialists, I still had no hearing in my left ear. The doctor told me that he did not know the reason I had lost my hearing. The doctor told me that since two courses of steroids had done nothing, I was unlikely to hear again. While he checked his schedule to make a follow-up apointment, I retired to the waiting room. Having a Bible with me and a brochure that tells you what passage to read each day of the year (meaning it was not my choice which pages I read), I opened to Exodus 4 and found Moses arguing with God:
10 Moses said to the Lord, “Pardon your servant, Lord. I have never
been eloquent, neither in the past nor since you have spoken to your
servant. I am slow of speech and tongue.”
11 The Lord said to him, “Who gave human beings their mouths? Who
makes them deaf or mute? Who gives them sight or makes them blind? Is
it not I, the Lord? 12 Now go; I will help you speak and will teach
you what to say.”
As far as I can tell, this is the only place in the whole Bible that directly says that God makes people deaf. As I read this passage, I understood that though the doctors did not know why I was deaf, God knew. More importantly, I knew that God understood my distress and by this "coincidence" proved to me that he was there with me. I left the doctor's office in peace. I did not take it as a promise that God would miraculously heal me. I did take it as a sign that he was real and he loved me.
Two days later, my hearing returned. Was it a miracle? Did my body naturally heal itself? Did the medicine take longer to work than expected? I can't answer those questions. All I know is that I believe that God hears our prayers and speaks to us in mysterious ways. I had two days of peace due to my faith that I would not have had if I merely waited those two days in fear and then got my hearing back. That IS a miracle.
What is the point of this story? Over the course of three decades as a Christian, I have amassed hundreds of small "coincidences" as well as more convincing experiences that reinforce my belief. I had none of these when I became a Christian, but have acquired them in the years since. My faith is logical, because to attribute a few of these happenings to chance is reasonable but to do so for all of them is unreasonable. It is more logical to believe that God exists. And because these happenings are so intertwined with the specific words in the Bible, each event expands the number of books or chapters in the Bible for which I have personal confirmation. That is because I believe that its words are both true and powerful - and that the power would not be there without the truth.
In the same way that my good experiences confirm the general truth of the Bible, my bad experiences have often exposed my misunderstanding of some of what it teaches. It goes both ways. Not every belief that I have is correct, and over time I replace the faulty ones that cause problems in my relatioships or other areas of life with better ones.