By "abiogenesis," I am assuming you are referring to the arisal of life from non-living matter through natural means. I think a big part of the misunderstanding and refusal to accept life emerging from non-life naturally is that most people seem to have an "either-or" view of life: either something is alive, or it isn't. This however ignores the broad range of stuff in between. For instance, there are a number of self-replicating molecules in nature. Can a mere molecule be said to be alive, just because it can make copies of itself? Unless you're an animist who believes everything is alive, the answer to that is probably no. But what if that molecule is a protein? Or a single strip of RNA? Or a strip of RNA encased in a protein coating? Now you're talking about a virus. From there it's not all that far to get to a simple single-celled organism, and from there....
Regarding the big bang theory (btw it's formally called Inflation Theory), its strength is that it fits in so beautifully with what we observe. Any good scientific theory makes testable predictions which can be confirmed or refuted through observations and/or experiments. In the case of Inflation Theory, it makes a number of predictions. It predicts that there should be a background of energy which cools down (redshifts) as the universe expands and ages; we have observed this as the microwave background radiation, and it is at just the temperature we would expect it to be at. The theory also predicts that our universe should have certain ratios of hydrogen to helium, lithium, and other heavier elements, and again, what we've observed dovetails with the theory. Ditto for the distribution pattern of matter throughout the universe.
For more on this, I very highly recommend "The Inflationary Universe" by Alan Guth, one of Inflation Theory's developers. This book does a good job of explaining in layman's terms not only what Inflationary Theory is, but just why it's our best theory in explaining our universe and what tests the theory had to go through to be accepted.
What might disprove a theory is if an observation is made or an experiment done which flies in the face of what the theory says is supposed to be. So for instance if we observed twice as much helium in the universe as we do now, that would be a problem for our current Inflationary Theory. When a theory runs into trouble like that, it must be either modified or chucked out the window. Either way, the process of scientific inquiry is to then find a theory which _does_ explain the observations and/or experimental results.
Science certainly doesn't have all the answers, at least not yet. If it did, then there'd be no more need to do any science, since we'd already know all there is to know. One must be careful, however, not to confuse the fact that science _doesn't_ have the answer to a problem with the assertion that it _cannot_. Science and religion usually come into conflict when one, usually religion, intrudes upon the other's territory. Religion is ok if you're talking about morality or church doctrine, but we need to realize that when it comes to offering verifiable claims about the world around us, that religion's explanations were essentially wild guesses by primitive peoples before the time of science, logic, experimentation, or rational thought. We now know that disease is not demonic possession, that epilepsy is entirely natural and not divine, so too with famines and tsunamis, and that the sun does not orbit the Earth. Every time human knowledge has advanced contrary to religious teachings, there has been conflict between religion and science, but every time, that conflict was unnecessary. So too are the current conflicts over evolution and the origins of our universe.