Are self-doubt and desperation a normal part of scientific writing?
I start with you related questions: 


  Can I still be a good researcher if I constantly doubt the quality of my work? 


You got it the wrong way. The real question is: Can I be a good researcher without constant doubt of the quality of my work? And the answer to this question is no. You should always question your work and and strive to make it better. Doubt is an integral part of research. But beware, that you should come out of this with "yes, my work matters/is of good quality/is relevant…" often enough either by your own conclusion of by feedback of others.


  Is there something that I can do to mitigate this part of the process?


Not sure, but you do not really need to. Please read this thread to see how to deal with insecurity and doubt as a researcher. You'll also find that the answer to


  Are self-doubt and desperation a normal part of scientific writing?


is most likely to be yes for most people regarding doubt and partly regarding desperation (assuming that "scientific writing" and "doing research" are quite similar in this respect). As Nate Eldredge notes, you should critically ask yourself if your self-doubt is close to be "pathological". I am not qualified in any way to give advice on how to diagnose yourself but you may want to read about the impostor syndrome.