I have no intention of arguing whether this faculty truly is a faculty, I simply accept that it exists and each one of us decides if this is true or not from a subjective point of view. The following brief comments explain, to some degree, my point of view based on my subjective experience. In many ways this opinion can only have relevance for me, but it might be of some assistance to you.
There are many ways of defining intuition and all are as true as they are false. All we might say with any safety is that there is an 'inner' way of gaining knowledge. This inner way can bypass the intellect wholly or in part, and is not dependent upon material facts. We have been conditioned through many generations that only the intellect and reason can be trusted, that this is the only way we might garner information and knowledge. This part of us is ruled by Mercury, also known as The Trickster, so can we really trust it? I would say that we cannot. The intellect serves a very useful purpose, of course, but it is only a way to understanding - wisdom as it was once called - it is not understanding in itself. The Sun is the centre of our universe, it is the Lord of the Heavens and is symbolic of the Creator and as such is the centre of all wisdom. In the human body, the Microcosm, it corresponds to the heart, the symbolic centre, and so it also represents that same centre of wisdom. So, the understanding of which I speak occurs there, in the heart, but how does this process occur and what must we do to stimulate it?
Astrology is a divinatory art and, if that is accepted, immediately implies another way of gathering information than by way of the intellect. In my opinion, this is indeed true, but not to the extent that it requires no input whatever from the intellect. When we say that we are studying astrology, we are training ourselves (please note the latter word) to be able to interpret the Word of God. We must train our intellects and, by so doing, we train our intuitive parts. This occurs almost automatically, because what we are studying is a symbolic language, a magical language. Astrological symbols are not merely shorthand expressions, but, in some senses, seem to have a life of their own. They will make changes to the practitioner who exposes him or herself to them and who attempts to plumb their depths. There are conditions, of course, and one of those is that the practitioner has a "fear of God" which can be interpreted in many ways. Fundamentally it means a respect for a power greater than our own which, in turn, implies humility.
We must study astrology and for that we need our intellects, we must learn all those correspondences and we must do so in an orderly manner. As we do this the intuition is placed in abeyance for a time, and this is how it should be. The intuition is known to be a slippery and often unreliable faculty because fancy or wishful thinking can disguise itself as intuition; it, too, needs to be trained. It needs to be trained to respond to certain stimuli and in our case it needs to respond to astrological symbolism. This happens if the student sincerely wants to learn and remains receptive to that learning. If the student manages to keep arrogance and pride at bay, gradually the intuition begins to respond and to assist. In the early days it might simply open the mind to possibilities, helping the student to remember. With time it will inform every delineation, but not in any startling way, but softly and quietly; it makes things very obvious. Nevertheless, the astrologer will be able to explain, using astrological symbolism, how they arrived at their answer.
You might notice that I haven't referred to any external agency in the gaining of this 'extra' information. There are no voices in your ear, no thunder and lightning, no angels, or demons for that matter, giving you messages; the process is internal - INtuition. It is achieved through hard work and honesty, there is no glamour and no mystery. If you learn this art properly and sincerely, you will train your intuition to behave likewise.
So to answer my own questions:
Do we use intuition in astrology? If so, should we use it?
Given the above-mentioned conditions, we can't help but use it, it is a natural development of the process of learning a magical art. However, the learning process, study, must come first for it to be reliable and sustainable.
How much do we use it?
This increasingly becomes impossible to answer, because as astrology becomes part of one's life, so too does the use of the intuition.
Can we use it on demand?
This is answered by the previous comment. We use it "on demand" every time we expose ourselves to the correct stimuli: the symbols and their correspondences.
What are we doing when we do use it, i.e. what is the process?
We are reaching the part of every human being that corresponds to the Divine, the part that can know everything. But we can't cheat this part of ourselves, the process has to be gone through properly, there are no shortcuts.
If there are any questions relating to my remarks, then please feel free to ask them.