Hm, interesting topic...
First, about the whole 1=1 thing, I did actually find a "proof" somewhere the ends with 1 not being equal to 1, but it involves dividing by zero (technically, having a denominator of "a - b" and then defining a = b, making the denominator a - a). So that might be what some of you are thinking of.
Back to the topic, though. My answers:
1. Truth is factual, impartial information, whether we can actually obtain that information or not. For instance, say someone decides to bury a box under his home. We have no way of knowing what's in the box short of either asking the person, or digging it up ourselves. That doesn't change the fact that whatever he put in the box, is what's in the box. Similarly, if we find a dinosaur skeleton, we can say for a fact that that dinosaur somehow died. We won't be able to tell how it died, though we can possibly make some guesses based on the condition the bones were in and how they were positioned. The "truth" in that case is how the dinosaur actually died, even though no one can actually say for certain what that is.
2. It is important if you make decisions based on those beliefs. For instance, let's say that my personal banker is a young-earth creationist (in other words, he believes that God created all life as it is now about 6000-10000 years ago, despite all evidence to the contrary). Now, being a banker, what his belief of the world's origins doesn't actually matter - as long as he can still do my banking, he can believe whatever he wants. However, if he were a high school biology teacher instead, his belief of how old the earth is and whether or not we all evolved would definitely impact how well he does his job. Thus, his beliefs being based on truth would be significantly important. Similarly, if he had decided back in September to sell off all of my assets because the would would end in December, that would definitely be a bad thing for me; again, he's letting his personal beliefs interfere with his decisions, which can have disastrous results.
3. The scientific method has proven itself to be the best way for us to determine whether something is true or not. In short, you simply observe something, try and predict what's causing it, determine a way to test whether your guess is correct or not, then test it. Most importantly however, you need to come up with a result that would prove your guess false. If you try to say that something is the way it is "because God made it that way", you need to come up with some result that would convince you that your answer is false before you test it; if you would always assume your guess is correct no matter what result you get, you're not actually testing anything.
That being said, sometimes there's just no way for us to know if something is true or not. In that case however, you should still try to get an answer that's as true as possible; throwing around "God did it" as a panacea for every single question you have does not actually help at all.