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.

I also know people that are enthusiastic throughout the entire process and love writing scientific papers.

That's great! I'm not sure I know anyone who is enthusiastic through the entire process. Having moments of doubt is pretty common, if not universal. The best thing you can do is accept that you're going to have moments of desperation (stage 4), while realizing that you have control over the way out.

People often think that they can't work until they feel better. As you describe it, you feel mildly optimistic and then you somehow get to work again. But often times it's the reverse. You get to work, and making progress makes you feel more confident and optimistic. So sit down, set yourself to finishing a small chunk of work, thank stage 4 for its constructive criticism, and show it the door.