I can't imagine that anyone sets out to solve a philosophical problem with "absolutely no psychological bias" concerning what the correct solution will look like. The degree to which I think I've already surmised "the answer" to a problem before getting down to the hard work of solving it depends on the particular problem. But I doubt I ever embark on finding a solution with no preconceived notion at all about the right answer.
I don't think this method counts as intellectually dishonest in general, and especially not in philosophy, where the success of one's solution depends entirely on the quality of one's argumentation, which is open for all to judge. Unlike empirical scientists, philosophical problem-solvers can't fake data. If a philosopher's proposed solution to a problem isn't clearly supported by the argumentation that he/she provides, anyone who reads the proposal is in a position to see that. This bracingly high intellectual standard is one of the main virtues of philosophy when it's done properly.
I don't know how prevalent this method is among professional philosophers, but for the reason just given it wouldn't bother me to learn that the method is applied universally.
I'm not sure why Tyson, Hawking, Krauss, Dawkins, Coyne, Feynman, et al., express so much contempt for philosophy. But my best guess is that they're ignorant -- unaware -- of what philosophy is when it's done well, perhaps because they received little or no academic training in philosophy when they were undergraduate students. (By the time they reached graduate school in the sciences, it may have been too late for them to get that training even if they had been interested in getting it.)
I don't think they're using different definitions, at least not systematically. Krauss does claim that physics has redefined the words "something" and "nothing," but I think he's deeply mistaken (see Question 4759). In general, I find that when non-philosophers, including scientists, reason about philosophical issues, they do so sloppily: making elementary mistakes in inference, conflating concepts that ought to be kept distinct, and so on. That's unfortunate but not surprising, since reasoning well about philosophical issues takes training: why shouldn't it? My sense is that their rigorous training was focused very narrowly on particular topics in physics or biology, and outside that narrow range they're no better reasoners than the average person -- and perhaps worse, because the deference they've been accorded as scientists has made them overconfident (hence insufficiently careful) about topics outside their specialties. Nor do I think that they're attacking just bad philosophy, because I'm not convinced that they can distinguish bad philosophy from good.
If so, then just who is the jury that decides? ...then just who is right?
As I see it, those two questions don't go hand-in-hand. Which side in a debate has the better reasons isn't something that a jury (in any sense of 'jury') can decide. It's not like legal guilt, which is something that a jury (or its equivalent) must decide and which isn't guaranteed to match what a fully informed and impartial observer would decide. No one is legally guilty unless a jury (or its equivalent) decides that he/she is. But one side in a debate can have the better reasons even if everyone in the audience judges otherwise. I think this point holds for debates in general, including philosophical debates. That's one reason why the oft-repeated "Who's to say?" is the wrong question to ask when debating an issue in ethics: no one's say-so is necessary or sufficient for truth in ethics or in philosophy more generally.
The idea that the consensus, or even the unanimity, of an audience doesn't determine the quality of a debater's reasons goes back to the ancient dispute between Plato and the sophists about the nature and value of philosophy and rhetoric. Much has been written about that topic, including much that's available online.
Can studying philosophy make one's life worse?
Certainly it can, just as working out to get fit can make one's life worse: one who works out to get fit can thereby tear a hamstring and become laid-up and miserable, or thereby suffer a heart attack, etc. Indeed, working out to get fit can make one's life worse overall, i.e., all things considered. Ditto for studying philosophy. There's no reason to think otherwise.
But studying philosophy, or working out to get fit, can also make one's life better, including better overall. For many people, myself included, studying philosophy has improved their lives overall (although I began studying philosophy because of an intense curiosity about the issues it covers rather than because of a conscious desire to make my life better by studying it). I doubt I can say anything more helpful without knowing more about your own encounters with philosophy. For instance, by "more critical of everything," do you mean "more intellectually vigilant about everything" or "more negative and cynical in my attitude toward everything"? I see the former as a benefit, even if the latter isn't. I'm also curious to know which philosophers you've been reading and which of them belong to the handful about whom you're not critical (in whichever sense of "critical"). Without knowing more about you, all I can say is that my own experience of studying philosophy has been positive overall, which is fortunate since I can't realistically see myself doing anything else with the bulk of my time. I'd urge you not to give up on philosophy. If you'd like to contact me, my email address appears on my homepage.
Speaking just for myself, I react much as I did in answering Question 4636 and Question 4759. For reasons that I hope those answers make obvious, I don't regard the dismissive remarks of Hawking, Krauss, Dawkins, Tyson, and their ilk as worth taking seriously. As far as I can tell, their remarks stem from simple ignorance of philosophy, often coupled with ineptitude at it.
Many decades ago, something very close to your question seems to have motivated philosophy professor Paul Arthur Schilpp to launch the multi-volume Library of Living Philosophers, which now contains more than 30 volumes. You can find out more at this link. I seem to recall reading an editor's introduction by Schilpp that explains in detail his motivation for launching the series. You might search for it.
To answer your last question first: I don't regard philosophical inquiry as a game. On the contrary, it may well be the most intellectually serious form of inquiry there is. What philosophical inquiry amounts to is itself a matter of philosophical controversy, but I'm inclined to say that philosophical inquiry consists in thinking as carefully as we can about the most general and most fundamental questions we can ask.
To your other questions:
1. The definition of "knowledge" -- more accurately, the analysis of the concept of knowledge -- has been an energetically disputed topic in philosophy for more than 50 years. Some philosophers defend a version of the "true, justified belief" analysis, with "justified" understood in a variety of ways. Others defend an analysis that doesn't require anything they're prepared to call "justification." Still others think that it's a mistake to try analyzing the concept of knowledge into "more basic" concepts. You'll find much useful information about all of this in the SEP entry on the analysis of knowledge, available at this link.
2. I don't believe that Descartes meant his "I think; therefore, I am" to be a definition of existence. Instead, it's supposed to be an indubitable piece of reasoning, or else an indubitable unitary thought, rather than a definition of a term or an analysis of a concept. Notice, by contrast, that "I am; therefore, I think" doesn't seem indubitable: at any rate, it's not valid reasoning, since it's possible for a person to exist (in a coma, perhaps) and not be thinking.
This is a terrific question/problem, and Stephen's response is a very good one. I merely want to point out that it's possible to have another kind of response to the situation you're confronting. I would characterize the situation as one where you realize that everything is ultimately connected to everything else, every belief has infinite implications that require exploring before they are rationally affirmed, to every position there is an objection, to every objection there is a response, and the whole process never ends ... Now if you believe that in order to know anything, or act in some rational way, the process of inquiry that produces that knowledge or action must be "completed", then you'll be in the bind you describe. But maybe THAT is the belief to be given up (and maybe the bind you describe is itself a key argument for giving up that belief). Instead recognize that deliberation and explanation must always come to a temporary end at some point -- and that you should always believe/act with the best set of principles that are available to you, with the information that is available to you, at the time of believing/acting, w/o pretense that the process is complete. Then, rather than feel frustrated, you might even feel exhilarated by realizing that the process of inquiry never ends: the world is infinitely richer, deeper, more interesting than we can possibly realize. (By way of rough analogy: if you find "life" interesting, beautiful, exhilirating, then when you discover that the number of possible life forms may be infinite, is that a source of frustration or exhilaration? Frustration if you believe that unless the process of cataloging life forms is complete then something is missing; exhilarating if you celebrate the infinite set of possibilities.)
Or from another direction. Suppose you realize that you have no better reason (ultimately) to get out of the bed in the morning than to stay in bed. If so, then that infinite process of deliberation is neutral with respect to whether you get out of bed. So don't bother undertaking it, at least not every morning. Instead do your ordinary, limited deliberation: "well sleeping is lovely, but so is keeping my job. So I better get out of bed." That is pretty darn good reasoning, if you ask me, even if it isn't "ultimate" or "completed" reasoning -- but it's also the only kind of reasoning that matters on a day-to-day basis. (And when you realize how awesome is the infinite set of deliberations that you could ultimately undertake, you might find it quite exhilarating to get out of bed -- because after you get off work today you can get home and do a little philosophy ....)