The idea of an "accident" is really a false one. On one level, yes, the universe is a chaotic place. At the subatomic level, you can't even really talk about single particles being in a specific place or moving at a specific speed, you can only discuss these particles in a statistical manner.
One of the biggest problems that theories like Big Bang cosmology and evolution have is that even some of those that accept them don't actually understand them. The Big Bang was not an explosion, it was an expansion of space from a very hot, dense state to one where it cooled. While the very initial moments after the Big Bang are extremely difficult to probe (and prior to the Planck Time, may even be impossible), but everything after that is largely explainable even by our still limited understanding of physical law. For instance, the creation of matter was not an accident, but simply part of a chain of events that lead from a high density, high temperature state to a cooler one. It wasn't a chance event, but rather inevitable via physical laws. There may be some aspects of the Big Bang that could have happened differently, and lead to a different universe (this is largely surmised from the notion that the decoupling of the basic forces might have been different, altering certain fundemental physical constants).
But the fundemental notion that the universe was once much hotter and denser is simply bourne out by the evidence; in particular nucleosynthesis (the ratio of the main elements hydrogen, helium and lithium to other elements), the cosmic background microwave radiation (a relic of the early universe that comes from every direction) and the red shift of distant galaxies (demonstrating that the universe is expanding).
As to evolution, in one respect it is a fact. Simply stated, evolution is change in the genetic makeup of a population over time. Even in asexually reproducing species, mistakes in transcription means that a descendant will never be identical to its parent. The large majority of these mistakes are, unlike the claims of Creationists like JWs, not delerious, but in fact neutral, having no effect one way or the other on the fitness of the organism. A minority will be delerious or beneficial.
The genetic distance between one generation and the next is going to be very small, but it will be there. While the fossil record has long demonstrated this in a somewhat inprecise way, the science of molecular genetics has not only rewritten the evolutionary book, it has confirmed the big picture that the fossil record showed us. All extant organisms that we have looked at all show the genetic heritage of a common ancestor. In fact, though we humans like to feel ourselves special, we are really genetically not very different from your average tuna. Most of the structures are there, the same basic vertebrate body form. Genetically we are closely aligned, but all life, whether it's us, sea squirts or some hardy bacteria living in a hot spring, fit within a nested hieararchy of living organisms.
Evolution is not an accidental process, though it is not one that goes in an particular direction. It's raw fuel is genetic diversity, but the environment puts heavy constraints on how variant any individual can be.
As to abiogenesis, it is still an area of a lot of speculation, and you won't find most biologists and chemists claiming anything more than that a certain process may have lead to life. But there are some tantalizing clues. The Urey-Miller experiment demonstrated that amino acids could be produced in an ancient reducing atmosphere like that of the early Earth's. But in some ways that great experiment has been overtaken by a few key discoveries. The first is that amino acids and other complex organic compounds are in fact relatively common, and found all over the universe; in dust clouds and even orbiting stars, so that it is conceivable that an early Earth was in fact given its organic molecules by comets, rather than necessarily having to be produced locally. The second key discovery is that the sun's energy may not have been responsible at all, and that complex chemistry can occur in deep sea vents, where high temperatures and raw materials would abound. In fact, this is suggested by the fact that the most primitive and ancient of organisms are found in just such places; deep sea vents and in hot springs, places where energy is plentiful.
