I also would like to thank RandFan not only for his willingness to do this but for the patience and kindness he has shown me over the past months. Thanks.
I'll attempt to rephrase to make sure I'm understanding you – hopefully I won't mess it up too much:
Issue 1: Mormonism can't be true because:
1. It is impossible to "know" something outside of your five senses because we have some examples of people who thought they knew something outside of their five senses and they were wrong.
2. Some people have wanted to believe something so much that they then start to believe it – a self fulfilling prophecy, a placebo effect, which con men can use for their own benefit.
3. Some people don't get answers to their prayers and it's blamed on the person not doing what they need to in order to get an answer instead of blaming it on God not existing.
4. Many people have received different answers that conflict with Mormonism.
5. The Mormon Church thinks that faith is knowledge but it is not and so they really don't know their religion is true, they just have faith that it is.
Hopefully the above statements are accurate and not straw men. Please correct me if I've summarized incorrectly or oversimplified. I'll attempt to address each item below.
It seems that a common thread, that you have expressed before, also runs through these assertions: An unprovable future benefit is not worth the risk of being deceived by others or self. I'll try to address this as I go.
1. You say that "it assumes that it is possible to correctly "know" without error the truth of a proposition innately and not by external sensory perception. We know that this is not true." You then give a few examples where people have had feelings about things, thinking that they knew it to be true but it wasn't. The fact that some people have been wrong is not proof that it is impossible to receive knowledge through other than the five senses. You cannot therefore say, "we know that this is not true." You can say, however, that, "we know that in these cases it has been shown that these people have been deceived by their feelings." This would not prove it is impossible to know something from outside of your five senses, but rather that one must keep in mind that self and group deception is a real possibility. A very good suggestion with which I happen to agree.
2. I agree that this occurs. I believe that the proper way to gain hope, belief, faith, and finally knowledge is through testimony and experience. For example, if you're in a large building with a bunch of strangers scattered all over in it in separate rooms and one stranger runs by yelling, "fire!" (testimony) you will choose to believe him or not based on your past experiences. If a different person then runs by yelling fire, you may then, with multiple witnesses' testimonies, believe enough to be moved to action. It's possible they are conspiring or both deceived – you don't know yet for sure. So you get up and start looking. At this point you have only belief or faith - no knowledge. You have seen no fire, smelled no smoke. (By my definition, faith is a belief that is sufficiently strong that it moves me to action. If I need to call it trust or confidence to avoid confusion, let me know.) You then go find the fire yourself based on where they said it was. Once you see the fire for yourself and feel its heat (experience) you now have knowledge through your senses - and so you run out of the building based on that knowledge. You could have also just run out of the building without going to find the fire yourself and you might have been luckily right or you might have been unluckily deceived. There was a series of testimonies and experiences that increased your belief and faith to the point that it moved you to find out firsthand - and only then did you gain any true knowledge of the reality of the fire. You took a risk and acted without a sure knowledge. Let's say that I decide to believe the first guy and let's assume he's a con man. Let's also assume that I want to believe him because I want an excuse to go outside (it's nice out today), so I just follow him outside and then he and his buddies proceed to relieve me of my wallet. Certainly that's possible, and in this world, perhaps it's quite common. That doesn't prove that there will never be a real fire or that I should assume that anyone yelling fire is a con man. I may require more witnesses before believing it the next time though. In fact, I may demand that I get some physical evidence the next time and I may benefit from that decision by not getting mugged or there may really be a fire next time and perish because of that decision. This is an example of taking a risk of being deceived because of the huge payoff of leaving when there really is a fire. I could risk burning with the building because I don't want to be deceived by con men, or I could take the risk and possibly get out alive. I think most of us would go check out if there was a fire just based on testimony of a couple of strangers. We would be acting on faith (not knowledge) to find out if what they were saying is true because if it were true, it really would be important to know and act accordingly.
3. This would be similar to having a friend tell me that if I were to mix vinegar, baking soda, red food coloring, and soap, it would make a fun volcano-like mess. If I mix all of these ingredients correctly, I'll get a great science fair project. If I mistakenly use bleached flour instead of baking soda, then the experiment will most likely fail. Is it really invalid to suggest that the experiment might not have been executed properly? I have a choice to make based on my belief of what happened (it's belief or faith because I don't yet have knowledge of why it didn't work) - try the experiment again, checking to see where I might have incorrectly followed the instructions, or I could walk away from it claiming that my friend was playing a joke on me. If I told the friend that it didn't work and that I used vinegar, "white powder", soap, and red food coloring, and the friend says, "white powder? Are you sure that was baking soda?" Is he being a jerk? Is he putting pressure on me or being demeaning to me? Or is he just making a suggestion of a possibility. Unfortunately, many people choose to be mean or self-righteous as they suggest possibilities, but that just proves that they are currently broken, nothing else.
4. You say "Many of these people have followed the admonition of James 1:5 and got a very different answer from Mormonism." Who did? What were their specific questions? I cite my own firsthand account of specifically praying to know if the Book of Mormon is true and receiving several spiritual confirmations over several years that it is true. I could be deceived, but since there is no proof that I am or am not deceived - either scenario is possible. I have also asked about a handful of other important things that I have also found to be true. I really don't know anything else in this realm for sure - for example, I really don't know for sure if Gordon B. Hinckley (current president of the Mormon church) is the current living true prophet of God - so I have to rely on faith or belief, not knowledge in those cases. The specificity of these questions and answers is very important. If someone else prayed specifically, "Is the Catholic church the only true church?" the same possibilities exist for their answer as far as being deceived or receiving an answer to such a specific question. I believe that no two truths will conflict and therefore if we both get an answer that the church we attend is the only church with authority from God, one or the other, or both of us are wrong. My experience in talking to hundreds of people from different religions is that many times the question they are asking (if they ask at all, which was very rare in my experience) is vague rather than specific - like feeling good while giving service to someone in a church event. Most people I talked to just went because their parents went or because they liked and agreed with it. Those were my reasons as a kid, until I became an atheist. I'm now LDS (Mormon) but not for the same reasons as before.
5. I think you might be referring to some members of the church rather than the church doctrines themselves. My understanding of these terms is that faith is just belief that is strong enough to move me to action – like faith that if I turn on a light switch I will receive light – my faith is strong enough that I am moved to action to walk across the room, stubbing my toe on the way, and turn the light switch on - I did all that without knowing for sure if the light would go on or not - but it was a risk I was willing to take. The switch may be broken and so it isn't knowledge since I really don't know if all of the connections are in place and that the power isn't turned off. If there is any part that I don't know then it is not knowledge, it is just faith. I can know that I turned it on yesterday and it came on, however. Yes, there may be Mormons who say "I know Joseph Smith is a prophet of God" when they really only just believe it to be true. I said that when I was a kid because I heard other people say it. I was incorrect to say it though – A more truthful statement would be, "I believe he is a prophet" unless there was some experience that led me to know it for certain. I agree that people often think they know more than they actually do. That's a human fault. Agreed. But seeing people exhibit that fault doesn't prove that someone can't actually know something is true for certain. And taking the risk, in faith, of turning on the light and perhaps stubbing my toe in the process is worth the payoff of having light in the room.
I think you have expressed very well the risks involved in relying on faith and experiences outside of the five-senses. You have shown that there are such things as false positives (like believing a con man). We could boil it down to the question: "Do I walk away from the experiment because I might be deceived by my friend or even myself, or do I continue to try it out because the payoff is potentially large." We take that risk without a sure knowledge in other areas of life - like the examples I have given above. So why not take a similar risk with finding out the truth about the existence of God?
I believe that God wants each one of us to individually seek and find Him and so I believe He has made it impossible for one of us to prove His existence to the rest of us. I believe that part of why we're here is for us to choose independently what we really want in life without having "dad" watching over us all the time, which might taint our choices - much like having a manager standing over me would prevent me from typing this right now.

Therefore, I believe that it is part of a bigger plan - that seeking and finding Him is subjective and non-transferable and that we can only know Him once we've truly independently chosen Him over the various alternatives here in life.
We have many testimonies of people who have seen God firsthand and have testimonies of people who have seen the miracles that prophets and Christ have performed. Some magicians or con men have performed convincing tricks and some people who are insane say that they have seen God. Disproving the magicians and con men is a big part of what these forums and this organization is all about. We can then either choose to assume that all miracles or witnesses are false or we can admit that it is possible that some might be true. There is no proof to the contrary and so I choose to leave the option open and have found tangible rewards from doing so - much like taking the risk of acting on testimony that I'm not sure of when the payoff is potentially big.
Are we willing to take the risk to find out if God exists? If not, why? And how does it differ from the many risks we take each day? I might be killed during my drive to work or my company might go bankrupt and I might not get a check even if I do make it there in one piece. I have no knowledge from my five senses and no way to prove that I will make it there in one piece or that I'll get a paycheck, but I take the risk in faith anyway. Why take the risk? Because I believe the potential benefit outweighs the risk. Taking a risk in faith for a potential benefit in the end. We do it every day.
P.S. I propose that we discuss a single item until we both agree to move on to the next single item. We can choose to move on leaving the item unresolved (agreeing to disagree) or resolved (finding some common ground). Again, thanks to RandFan and the moderators for giving us the opportunity to do this.
P.P.S. Nope. Mine wasn't pithy either. Maybe next time.
