"If a man finds a woman beautiful, but his friend does not, does that mean the woman is not beautiful." No. For Kant, if a man finds a woman beautiful then he puts her on a pedestal and demands that others see her as beautiful too. (This of course makes one think of issues of sexism, but for now just think of the situation as no different from when a woman finds a man is beautiful.) I think Kant believed that if there is a disagreement then one of the two is not being disinterested...is thinking of the issue in terms of practicality, morality or cognition. This raises another issue. What if the man finds his girlfriend beautiful and his friend does not. He then is being "interested" since he really does care that she exists, whereas his friend might well be disinterested. In fact, the man is more likely to find her beautiful because he likes her, whereas the friend is more objective. But then, the beauty of the woman is more important to the man who finds her beautiful than it is to the friend: so shouldn't her have some priority here? This is where Kant's thinking may face a problem.
"When talking about dependent beauty is Kant implying that the beauty of man is not (and could not be) as great as that of a flower since he thinks that it presupposes a concept of a man's end or purpose?" This is a difficult question. Kant does not actually say that free beauty is more beautiful than dependent beauty, although we tend to get that impression. Actually I think that in the end dependent beauty is more important for Kant than free beauty: the beauty of fine art, for example, is dependent beauty. Of course it is strange to us today to think of there being something that defines what a man has to be and thus determines the extent to which a particular man is perfect as a specimen.
"What is the form of finality? It is in the object, but has no purpose?" Kant believes that "Beauty is the form of finality in an object, so far as perceived in it apart from the representation of an end." So, when we see something as beautiful we notice that it has the look of a purpose, i.e. it looks designed, but we do not have the explicit thought of what its purpose might be. This happens with two kinds of object. The first kind is objects that actually have a purpose, or have elements that have a purpose. A flower for example has stamen. These have a purpose in reproduction. So when we look at the flower we are supposed not to think of that actual purpose (even the Botanist!) but only focus on the designed look of the flower. The other kind of object is something that looks designed but there is no clear reason to believe that this design serves a function. We can also find that beautiful.
"Is it possible for an object to be both beautiful and universally agreeable, and if that were possible would that be something like the holy grail of objects?" It would be possible for something to be both beautiful and agreeable, for Kant. It would be beautiful because its look causes the cognitive faculties of imagination and understanding to harmoniously engage in free play (in response to the look of purpose in the object). It would be agreeable because it pleases the senses. So a painting could be beautiful because of its composition and agreeable because of its colors. Universality is a different issue: Kant believed that all humans have imagination and understanding that are basically the same (he seems not to have any reason to believe this). He believed that when it comes to our sense organs, however, each of us is different. So universality would be much more difficult when it comes to the agreeable.
"Doesn't everyone have their own definition of beauty that doesn't necessarily agree with everyone, and hence they can call something beautiful without having to please everyone?" Kant does not believe that you have to please everyone when you call something beautiful or even that the object you call beautiful has to please everyone. It has to please everyone who is looking at it in a disinterested way and focusing on the designed or design-like look that can cause the mind to go into free play. Whether everyone has their own definition of beauty is another question. Very few people have their own explicit definition of beauty, but it is true that everyone has a different set of things they consider or have considered beautiful, and you might argue that this is based on their having a different implicit definition of beauty. I am not sure everyone does have an implicit definition since even an implicit definition requires some rules, and it may just be that the sets of things considered beautiful by many people are just arbitrary sets. Perhaps you could say that everyone has at least some general idea of what they consider to be beautiful. In any case, Kant would say that beauty does not depend on a definition. It depends on how our faculties respond to a thing that has a look of purpose when we view it in a disinterested fashion.
"If something pleases everybody, is there a need for a standard of taste?" You probably still do since you do not know whether it pleases everyone because it is agreeable, morally good, or beautiful. You need a standard of taste to distinguish these.
"Is there anything that, when looked at by anybody, would give them a jaw-dropping experience of awe and appreciation?" This question may presuppose another definition of "beauty" than Kant would accept, i.e. one that entails jaw-droppingness. Kant would probably reply that there are lots of things that that would cause anybody and everybody to be pleased in appreciating it aesthetically as long as they approach it with disinterestedness and focused on the form of finality in the object in such a way as to cause imagination and understanding to go into free play.
"A child may not have background scientific knowledge on a subject, allowing him or her to easily have an aesthetic experience on Kant's view. Can one really set aside background knowledge about a subject when viewing it aesthetically?" I think we can set aside things that we know simply by not thinking about them. So the botanist could just not think about the function of the stamen when appreciating a flower. It might take some training however to do this. The "child" question is interesting since the child could also be disinterested in the sense of not thinking about moral issues and not being interested in such practical issues as monetary or sexual reward. Moreover, the imagination and understanding of a child are constantly in free play: that is what we get the word "play" from: from what a child does. So it would seem that contrary to Hume, who stresses experience, the ideal Kantian judge would be a child, or a playful disinterested child who has an eye for the look of design. That seems odd.