Should I start writing a paper early or after all research is finished?
The concept of "finished" is problematic when it comes to research.  I think that the same quote applies as for art: research is never finished, it is only abandoned.

Less poetically and more pragmatically, it is often only in the process of writing that certain critical aspects of the work become apparent.  When a person is in the midst of working on a project, they tend to get very close to the material and begin to take as clear and obvious things that are very much not so for others who are not so deeply involved.  Writing one's work up in a scientific paper forces one to step back and build those gone-implicit arguments from the ground up (or at least it does if you are writing well).  

This often leads to discovering unexpected problems, which lead to new literature searches, new theorems, new experiments, and even whole new perspectives.  I have had nearly the entirety of a paper change out from under me as we wrote it and revised it, and the work became much better as a result.

So, to return to your question, of when to begin writing up a paper.  My advice and experience is this: begin writing when you think you have achieved the key results that you want to build the paper around.  As you begin to do so, you will likely discover gaps that need to be filled in, which will shift how you write the paper, etc.  When the process converges, you know you've got a good paper on your hands, and it is ready to submit into the tender mercies of your dreaded peers.

Don't let yourself move forward with the research, though, to try to achieve the next key result.  It's fun and exciting to do new things, but you must also have the discipline to cross the Is, dot the Ts, and observe the little things that need to be corrected and might otherwise escape your notice.