I do research for my thesis. The thesis is about cats. So I do research about cats.

For your first question: I think you should say: a research on some subject. for example: I'm conducting a research on Algorithm complexity analysis. For you second question, the difference between Research and search, Research means To study something thoroughly, a kind of Scholarly or scientific investigation and to search means to look over carefully in order to find something.

If you are trying to find causes you can also use : research into I conduct research into the causes of Alzheimer's disease.