Questioning faith for some reason has a connotation as being bad. I think questioning faith is an essential part of existence and an essential part of having any true faith. So, I encourage the questioning of one's faith. Of course, I believe it is also necessary that, in doing so, one is sure to get arguments from BOTH sides. I think that the act questioning faith the way most people undertake it can be more accurately described as researching all the counter-arguments to faith. Since I don't believe that one can actually have faith without questioning it first, simply researching all the counter-arguments is not questioning faith but, rather, solidifying lack of faith.
So, how does one question faith in a nutshell? Hell if I know
But there are some things i'm pretty certain about. Clearly, one does need to expose oneself to the counter-arguments of any faith. Exposure and consideration of these is necessary for developing what I would call actual faith. One also must discover the arguments and counter counter-arguments of faith. There are two important points of this though. Obviously, one should find out what the "official" arguments are. This is not as simple as it seems though. For example, if one is questioning Judaism, there are many different "official" arguments depending on whether it is Orthodox Judaism, Hasidic Judaism, Reform Judaism, and so on. But that is not all. This is where the second part comes in: one needs to look into the other theological trends of the faith as well. For example, in the case of Catholicism, there are certain contemporary theological trends at work right now that one cannot find in the "official" arguments. It's not so much that the "official" arguments are wrong, or that contemporary Catholic theology is right, but that theology is a constantly evolving thing. With new understandings of life and existence that come in the world through, for example, science, so too must theology evolve to reflect our growing understanding of ourselves. Clearly, it takes longer for something like this to occur in "official" teaching than in contemporary thought. One must simply keep in mind that it is, for lack of a better term, the "cutting edge" of theology.
An example: I am Catholic and I have come to be fairly secure in a belief that the sacrament of baptism is not a mystical "instant cleansing" of some ailment that is passed down from person to person since the first humans (Catholics do not reject evolution and I find it impossible not to accept it). Rather, through discussions with some Catholic theologians (it helps that I go to a Catholic university of course, but that doesn't make it impossible for those who don't) and the reading of books by various Catholic priests on the more contemporary end of Catholic theological thought, I have come to the conclusion that the sacrament of baptism is a largely symbolic and beautiful initiation rite into the Church. There are already hints towards this line of thought in modern, post-Vatican II "official" Catholic teaching. For example, the belief that baptism is not absolutely necessary for one to find salvation - whatever that may be described as. However, IMO at least, it takes time for these thoughts to be taken to their logical conclusions in the "official" circles of theology. It's not that I reject these "official" teachings, but more accurately that I think there is a better way to describe them. For example, despite the recognition since Vatican II that baptism is not something that is absolutely 100% necessary, the Church's definition of Original Sin remains the same - seemingly in contradiction with this stance on baptism. (Incidentally, I think original sin can be more accurately described as simply the inherent flaws that make us human - the drive for power, greed, selfishnes, etc. Indeed, these ARE things passed down from generation to generation, but they are not things which can be instantly cleansed away or things whose presence is an instant preventor to being a good person and connecting with God.)
Based on the (very) short description above, you can probably guess that I don't believe one ever stops questioning faith. Questioning faith is a lifelong process to me, and in my opinion it is the only way it can be meaningful. Being set in one's ways is detrimental to mental and spiritual health. If one chooses to be Muslim and ceases to critically contemplate Islam in relation to the world in which they live, it becomes not faith but comfort. Rather than having faith, one is instead simply sticking to what they know. Likewise, if one chooses at some point in their life to become atheist or agnostic but ceases to be critical of atheism or ceases to be considerate of other possibilities, one becomes stagnant and is no longer actively rejecting anything but, rather, simply comfortable with not challenging themselves.
Any kind of serious quest to find or reject faith of any sort is, I believe, a lifelong process which involves intense contemplation, discussion, and reading. Anything less I think is a disservice to oneself.
To speak more directly to your own description, a few books I am quite fond of in addressing various concepts in Catholicism are
The Great Mysteries,
Pierre Teilhard De Chardin: Writings, and
Jesus in the New Universe Story.