Welcome to BiologyBeta.SE!
On stackExchange sites, an OP should always limit his/her post to one question and should make his/her question is not too broad. But before you split up your question I invite you to have a look to some other post that will give you some answers. For this reason your question will very likely get closed as too broad. But I am giving some indices below. Also your post is a bit poorly written. This also lead people to close posts. Please come back with better posts, show some efforts in seeking for some information and it will be my pleasure to help you!
Origin of life
There are already a bunch of posts on the origin of life for example. Note that the origin of life is not a field that is part of evolutionary biology as evolutionary biology only explain the evolution of life and not its origin. As a mathematician you may prefer saying something like the domain of definition of evolutionary biology does not include the origin of life just like the shannon's theory of information says nothing about the euclidian geometry.
What i understand till now is that the "improvement" of the dna is a purely random development and the Selection criteria if the improvement was not negative is by surviving
Hum… that's sounds very weird. If I didn't know I would hardly recognize that you talk about evolution. You should have a look to some introduction to the field. Also, you should not confound the concepts of evolution, natural selection and mutations that I guess you confuse. A good, extremely simplistic view of evolution is that mutations occurs and create variance in the population. This traits are heritable. The variants that replicate the best increase in frequency in the population while the other variants will decrease in frequency up to the point to disappear. This is called natural selection.
Consider a pre-bird without wings, to get wings it has to have interstage with some useless pre-wings and then it has to learn how to flight
You also seem to ask a question about how some traits may come to exist while what seem to be necessary intermediate would not be beneficial. This question has been answered probably a thousands of time as it is a common creationist argument.
Because your questions really sounds like creationist arguments I am a bit afraid that you are not trying to understand evolution but you wondering whether evolution does occur or not.
Note also that evolutionary biology is a very big field. You cannot expect to have a more or less good understanding of the whole science of evolutionary biology in less than a year. But if you are trying to just understand the very basic, sufficiently to persuade you that evolution does occur, then few weeks will probably be enough if you show some effort
Please don't forget the questions you may have. Seek by yourself for more knowledge and if you can't find something, then you're really welcome to come back to this site with an accurate question.
Have a look to the definition of scientific theory. Also, please read @Chris comment below.
You misunderstand the meaning of "direct observation" that canadianer used. Observations we do in evolutionary biology are just as valid as those that has been made to confirm the theory of gravity. The only point concerns the easiness of understand these observations for non-literate in the field.
Yes, you are totally right. Just like the theory of gravity, the theory of evolution does not explain everything. It only applies to a subset of the phenomenons existing on earth. The theory of evolution is true, we know for a fact that evolution occurs but our theory is not global enough to explain things like the big bang or the surface tension and will never do so. The theory of evolution will never prove or disprove the existence of God as well (supposing you're concerned about that) because again it is not within its scope and will never be. However, the sociological impact of this theory tells much about the human behavior and our difficulties to accept something we know to be true if it contradicts some pre-establish believes even when these pre-established believes are not based on any logical developments.
Your question has been closed because it is too broad. The issue also is that your questions concern much more than evolutionary biology and biology. It also concerns the very nature of science and this is a common reason why creationist missundestand evolution. They don't (only) missunderstand evolution but they missunderstand what science is.